# Isqrt (integer square root) of X

Isqrt (integer square root) of X
You are encouraged to solve this task according to the task description, using any language you may know.

Sometimes a function is needed to find the integer square root of   X,   where   X   can be a real non─negative number.

Often   X   is actually a non─negative integer.

For the purposes of this task,   X   can be an integer or a real number,   but if it simplifies things in your computer programming language,   assume it's an integer.

One of the most common uses of   `Isqrt`   is in the division of an integer by all factors   (or primes)   up to the    X    of that integer,   either to find the factors of that integer,   or to determine primality.

An alternative method for finding the   `Isqrt`   of a number is to calculate:       floor( sqrt(X) )

•   where   sqrt    is the   square root   function for non─negative real numbers,   and
•   where   floor   is the   floor   function for real numbers.

If the hardware supports the computation of (real) square roots,   the above method might be a faster method for small numbers that don't have very many significant (decimal) digits.

However, floating point arithmetic is limited in the number of   (binary or decimal)   digits that it can support.

For this task, the integer square root of a non─negative number will be computed using a version of   quadratic residue,   which has the advantage that no   floating point   calculations are used,   only integer arithmetic.

Furthermore, the two divisions can be performed by bit shifting,   and the one multiplication can also be be performed by bit shifting or additions.

The disadvantage is the limitation of the size of the largest integer that a particular computer programming language can support.

Pseudo─code of a procedure for finding the integer square root of   X       (all variables are integers):

```         q ◄── 1                                /*initialize  Q  to unity.  */
/*find a power of 4 that's greater than X.*/
perform  while q <= x         /*perform while  Q <= X.    */
q ◄── q * 4                   /*multiply  Q  by  four.    */
end  /*perform*/
/*Q  is now greater than  X.*/
z ◄── x                                /*set  Z  to the value of X.*/
r ◄── 0                                /*initialize  R  to zero.   */
perform  while q > 1          /*perform while  Q > unity. */
q ◄── q ÷ 4                   /*integer divide by  four.  */
t ◄── z - r - q               /*compute value of  T.      */
r ◄── r ÷ 2                   /*integer divide by  two.   */
if t >= 0  then do
z ◄── t       /*set  Z  to value of  T.   */
r ◄── r + q   /*compute new value of  R.  */
end
end  /*perform*/
/*R  is now the  Isqrt(X).  */

/* Sidenote: Also, Z is now the remainder after square root (i.e.  */
/*           R^2 + Z = X, so if Z = 0 then X is a perfect square). */
```

Another version for the (above)   1st   perform   is:

```                  perform  until q > X          /*perform until  Q > X.     */
q ◄── q * 4                   /*multiply  Q  by  four.    */
end  /*perform*/
```

Integer square roots of some values:

```Isqrt( 0)  is   0               Isqrt(60)  is  7                Isqrt( 99)  is   9
Isqrt( 1)  is   1               Isqrt(61)  is  7                Isqrt(100)  is  10
Isqrt( 2)  is   1               Isqrt(62)  is  7                Isqrt(102)  is  10
Isqrt( 3)  is   1               Isqrt(63)  is  7
Isqrt( 4)  is   2               Isqrt(64)  is  8                Isqet(120)  is  10
Isqrt( 5)  is   2               Isqrt(65)  is  8                Isqrt(121)  is  11
Isqrt( 6)  is   2               Isqrt(66)  is  8                Isqrt(122)  is  11
Isqrt( 7)  is   2               Isqrt(67)  is  8
Isqrt( 8)  is   2               Isqrt(68)  is  8                Isqrt(143)  is  11
Isqrt( 9)  is   3               Isqrt(69)  is  8                Isqrt(144)  is  12
Isqrt(10)  is   3               Isqrt(70)  is  8                Isqrt(145)  is  12
```

•   the `Isqrt` of the     integers     from     0 ───► 65    (inclusive), shown in a horizontal format.
•   the `Isqrt` of the   odd powers  from   71 ───► 773   (inclusive), shown in a   vertical   format.
•   use commas in the displaying of larger numbers.

You can show more numbers for the 2nd requirement if the displays fits on one screen on Rosetta Code.
If your computer programming language only supports smaller integers,   show what you can.

## 11l

Translation of: D
```F commatize(number, step = 3, sep = ‘,’)
V s = reversed(String(number))
String r = s[0]
L(i) 1 .< s.len
I i % step == 0
r ‘’= sep
r ‘’= s[i]
R reversed(r)

F isqrt(BigInt x)
assert(x >= 0)

V q = BigInt(1)
L q <= x
q *= 4

V z = x
V r = BigInt(0)
L q > 1
q I/= 4
V t = z - r - q
r I/= 2
I t >= 0
z = t
r += q

R r

print(‘The integer square root of integers from 0 to 65 are:’)
L(i) 66
print(isqrt(BigInt(i)), end' ‘ ’)
print()

print(‘The integer square roots of powers of 7 from 7^1 up to 7^73 are:’)
print(‘power                                    7 ^ power                                                 integer square root’)
print(‘----- --------------------------------------------------------------------------------- -----------------------------------------’)
V pow7 = BigInt(7)
V bi49 = BigInt(49)
L(i) (1..73).step(2)
print(‘#2 #84 #41’.format(i, commatize(pow7), commatize(isqrt(pow7))))
pow7 *= bi49```
Output:
```The integer square root of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of powers of 7 from 7^1 up to 7^73 are:
power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
```

```with Ada.Text_Io;

procedure Integer_Square_Root is

function Isqrt (X : Big_Integer) return Big_Integer is
Q       : Big_Integer := 1;
Z, T, R : Big_Integer;
begin
while Q <= X loop
Q := Q * 4;
end loop;
Z := X;
R := 0;
while Q > 1 loop
Q := Q / 4;
T := Z - R - Q;
R := R / 2;
if T >= 0 then
Z := T;
R := R + Q;
end if;
end loop;
return R;
end Isqrt;

function Commatize (N : Big_Integer; Width : Positive) return String is
S     : constant String := To_String (N, Width);
Image : String (1 .. Width + Width / 3) := (others => ' ');
Pos   : Natural := Image'Last;
begin
for I in S'Range loop
Image (Pos) := S (S'Last - I + S'First);
exit when Image (Pos) = ' ';
Pos := Pos - 1;
if I mod 3 = 0 and S (S'Last - I + S'First - 1) /= ' ' then
Image (Pos) := ''';
Pos := Pos - 1;
end if;
end loop;
return Image;
end Commatize;

type Mode_Kind is (Tens, Ones, Spacer, Result);
begin
Put_Line ("Integer square roots of integers 0 .. 65:");
for Mode in Mode_Kind loop
for N in 0 .. 65 loop
case Mode is
when Tens   =>  Put ((if N / 10 = 0
then "  "
else Natural'Image (N / 10)));
when Ones   =>  Put (Natural'Image (N mod 10));
when Spacer =>  Put ("--");
when Result =>  Put (To_String (Isqrt (To_Big_Integer (N))));
end case;
end loop;
New_Line;
end loop;
New_Line;

declare
package Integer_Io is new Ada.Text_Io.Integer_Io (Natural);
N    : Integer    := 1;
P, R : Big_Integer;
begin
Put_Line ("|  N|" & 80 * " " & "7**N|" & 30 * " " & "isqrt (7**N)|");
Put_Line (133 * "=");
loop
P := 7**N;
R := Isqrt (P);
Put ("|");  Integer_Io.Put (N, Width => 3);
Put ("|");  Put (Commatize (P, Width => 63));
Put ("|");  Put (Commatize (R, Width => 32));
Put ("|");  New_Line;
exit when N >= 73;
N := N + 2;
end loop;
Put_Line (133 * "=");
end;

end Integer_Square_Root;
```
Output:
```Integer square roots of integers 0 .. 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
------------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

|  N|                                                                                7**N|                              isqrt (7**N)|
=====================================================================================================================================
|  1|                                                                                   7|                                         2|
|  3|                                                                                 343|                                        18|
|  5|                                                                              16'807|                                       129|
|  7|                                                                             823'543|                                       907|
|  9|                                                                          40'353'607|                                     6'352|
| 11|                                                                       1'977'326'743|                                    44'467|
| 13|                                                                      96'889'010'407|                                   311'269|
| 15|                                                                   4'747'561'509'943|                                 2'178'889|
| 17|                                                                 232'630'513'987'207|                                15'252'229|
| 19|                                                              11'398'895'185'373'143|                               106'765'608|
| 21|                                                             558'545'864'083'284'007|                               747'359'260|
| 23|                                                          27'368'747'340'080'916'343|                             5'231'514'822|
| 25|                                                       1'341'068'619'663'964'900'807|                            36'620'603'758|
| 27|                                                      65'712'362'363'534'280'139'543|                           256'344'226'312|
| 29|                                                   3'219'905'755'813'179'726'837'607|                         1'794'409'584'184|
| 31|                                                 157'775'382'034'845'806'615'042'743|                        12'560'867'089'291|
| 33|                                               7'730'993'719'707'444'524'137'094'407|                        87'926'069'625'040|
| 35|                                             378'818'692'265'664'781'682'717'625'943|                       615'482'487'375'282|
| 37|                                          18'562'115'921'017'574'302'453'163'671'207|                     4'308'377'411'626'977|
| 39|                                         909'543'680'129'861'140'820'205'019'889'143|                    30'158'641'881'388'842|
| 41|                                      44'567'640'326'363'195'900'190'045'974'568'007|                   211'110'493'169'721'897|
| 43|                                   2'183'814'375'991'796'599'109'312'252'753'832'343|                 1'477'773'452'188'053'281|
| 45|                                 107'006'904'423'598'033'356'356'300'384'937'784'807|                10'344'414'165'316'372'973|
| 47|                               5'243'338'316'756'303'634'461'458'718'861'951'455'543|                72'410'899'157'214'610'812|
| 49|                             256'923'577'521'058'878'088'611'477'224'235'621'321'607|               506'876'294'100'502'275'687|
| 51|                          12'589'255'298'531'885'026'341'962'383'987'545'444'758'743|             3'548'134'058'703'515'929'815|
| 53|                         616'873'509'628'062'366'290'756'156'815'389'726'793'178'407|            24'836'938'410'924'611'508'707|
| 55|                      30'226'801'971'775'055'948'247'051'683'954'096'612'865'741'943|           173'858'568'876'472'280'560'953|
| 57|                   1'481'113'296'616'977'741'464'105'532'513'750'734'030'421'355'207|         1'217'009'982'135'305'963'926'677|
| 59|                  72'574'551'534'231'909'331'741'171'093'173'785'967'490'646'405'143|         8'519'069'874'947'141'747'486'745|
| 61|               3'556'153'025'177'363'557'255'317'383'565'515'512'407'041'673'852'007|        59'633'489'124'629'992'232'407'216|
| 63|             174'251'498'233'690'814'305'510'551'794'710'260'107'945'042'018'748'343|       417'434'423'872'409'945'626'850'517|
| 65|           8'538'323'413'450'849'900'970'017'037'940'802'745'289'307'058'918'668'807|     2'922'040'967'106'869'619'387'953'625|
| 67|         418'377'847'259'091'645'147'530'834'859'099'334'519'176'045'887'014'771'543|    20'454'286'769'748'087'335'715'675'381|
| 69|      20'500'514'515'695'490'612'229'010'908'095'867'391'439'626'248'463'723'805'607|   143'180'007'388'236'611'350'009'727'669|
| 71|   1'004'525'211'269'079'039'999'221'534'496'697'502'180'541'686'174'722'466'474'743| 1'002'260'051'717'656'279'450'068'093'686|
| 73|  49'221'735'352'184'872'959'961'855'190'338'177'606'846'542'622'561'400'857'262'407| 7'015'820'362'023'593'956'150'476'655'802|
=====================================================================================================================================
```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

```BEGIN # Integer square roots #
PR precision 200 PR
# returns the integer square root of x; x must be >= 0                   #
PROC isqrt = ( LONG LONG INT x )LONG LONG INT:
IF   x < 0 THEN print( ( "Negative number in isqrt", newline ) );stop
ELIF x < 2 THEN x
ELSE
# x is greater than 1                                            #
# find a power of 4 that's greater than x                        #
LONG LONG INT q := 1;
WHILE q <= x DO q *:= 4 OD;
# find the root                                                  #
LONG LONG INT z := x;
LONG LONG INT r := 0;
WHILE q > 1 DO
q OVERAB 4;
LONG LONG INT t = z - r - q;
r OVERAB 2;
IF t >= 0 THEN
z  := t;
r +:= q
FI
OD;
r
FI; # isqrt #
# returns a string representation of n with commas                       #
PROC commatise = ( LONG LONG INT n )STRING:
BEGIN
STRING result      := "";
STRING unformatted  = whole( n, 0 );
INT    ch count    := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF   ch count <= 2 THEN ch count +:= 1
ELSE                    ch count  := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END; # commatise #
# left-pads a string to at least n characters                            #
PROC pad left = ( STRING s, INT n )STRING:
BEGIN
STRING result := s;
WHILE ( UPB result - LWB result ) + 1 < n DO " " +=: result OD;
result
print( ( "Integer square roots of 0..65", newline ) );
FOR i FROM 0 TO 65 DO print( ( " ", whole( isqrt( i ), 0 ) ) ) OD;
print( ( newline ) );
# integer square roots of odd powers of 7                                #
print( ( "Integer square roots of 7^n", newline ) );
print( ( " n|", pad left( "7^n", 82 ), "|", pad left( "isqrt(7^n)", 42 ), newline ) );
LONG LONG INT p7 := 7;
FOR p BY 2 TO 73 DO
print( ( whole( p, -2 )
, "|"
, pad left( commatise(        p7   ), 82 )
, "|"
, pad left( commatise( isqrt( p7 ) ), 42 )
, newline
)
);
p7 *:= 49
OD
END```
Output:
```Integer square roots of 0..65
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Integer square roots of 7^n
n|                                                                               7^n|                                isqrt(7^n)
1|                                                                                 7|                                         2
3|                                                                               343|                                        18
5|                                                                            16,807|                                       129
7|                                                                           823,543|                                       907
9|                                                                        40,353,607|                                     6,352
11|                                                                     1,977,326,743|                                    44,467
13|                                                                    96,889,010,407|                                   311,269
15|                                                                 4,747,561,509,943|                                 2,178,889
17|                                                               232,630,513,987,207|                                15,252,229
19|                                                            11,398,895,185,373,143|                               106,765,608
21|                                                           558,545,864,083,284,007|                               747,359,260
23|                                                        27,368,747,340,080,916,343|                             5,231,514,822
25|                                                     1,341,068,619,663,964,900,807|                            36,620,603,758
27|                                                    65,712,362,363,534,280,139,543|                           256,344,226,312
29|                                                 3,219,905,755,813,179,726,837,607|                         1,794,409,584,184
31|                                               157,775,382,034,845,806,615,042,743|                        12,560,867,089,291
33|                                             7,730,993,719,707,444,524,137,094,407|                        87,926,069,625,040
35|                                           378,818,692,265,664,781,682,717,625,943|                       615,482,487,375,282
37|                                        18,562,115,921,017,574,302,453,163,671,207|                     4,308,377,411,626,977
39|                                       909,543,680,129,861,140,820,205,019,889,143|                    30,158,641,881,388,842
41|                                    44,567,640,326,363,195,900,190,045,974,568,007|                   211,110,493,169,721,897
43|                                 2,183,814,375,991,796,599,109,312,252,753,832,343|                 1,477,773,452,188,053,281
45|                               107,006,904,423,598,033,356,356,300,384,937,784,807|                10,344,414,165,316,372,973
47|                             5,243,338,316,756,303,634,461,458,718,861,951,455,543|                72,410,899,157,214,610,812
49|                           256,923,577,521,058,878,088,611,477,224,235,621,321,607|               506,876,294,100,502,275,687
51|                        12,589,255,298,531,885,026,341,962,383,987,545,444,758,743|             3,548,134,058,703,515,929,815
53|                       616,873,509,628,062,366,290,756,156,815,389,726,793,178,407|            24,836,938,410,924,611,508,707
55|                    30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943|           173,858,568,876,472,280,560,953
57|                 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207|         1,217,009,982,135,305,963,926,677
59|                72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143|         8,519,069,874,947,141,747,486,745
61|             3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007|        59,633,489,124,629,992,232,407,216
63|           174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343|       417,434,423,872,409,945,626,850,517
65|         8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807|     2,922,040,967,106,869,619,387,953,625
67|       418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543|    20,454,286,769,748,087,335,715,675,381
69|    20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607|   143,180,007,388,236,611,350,009,727,669
71| 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743| 1,002,260,051,717,656,279,450,068,093,686
73|49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407| 7,015,820,362,023,593,956,150,476,655,802
```

## ALGOL W

Algol W integers are restricted to signed 32-bit, so only the roots of the powers of 7 up to 7^9 are shown (7^11 will fit in 32-bits but the smallest power of 4 higher than 7^11 will overflow).

```begin % Integer square roots by quadratic residue                            %
% returns the integer square root of x - x must be >= 0                  %
integer procedure iSqrt ( integer value x ) ;
if      x < 0 then begin assert x >= 0; 0 end
else if x < 2 then x
else begin
% x is greater than 1                                            %
integer q, r, t, z;
% find a power of 4 that's greater than x                        %
q := 1;
while q <= x do q := q * 4;
% find the root                                                  %
z := x;
r := 0;
while q > 1 do begin
q := q div 4;
t := z - r - q;
r := r div 2;
if t >= 0 then begin
z := t;
r := r + q
end if_t_ge_0
end while_q_gt_1 ;
r
end isqrt;
% writes n in 14 character positions with separator commas               %
procedure writeonWithCommas ( integer value n ) ;
begin
string(10) decDigits;
string(14) r;
integer    v, cPos, dCount;
decDigits    := "0123456789";
v            := abs n;
r            := " ";
r( 13 // 1 ) := decDigits( v rem 10 // 1 );
v            := v div 10;
cPos         := 12;
dCount       := 1;
while cPos > 0 and v > 0 do begin
r( cPos // 1 ) := decDigits( v rem 10 // 1 );
v      :=  v div 10;
cPos   := cPos - 1;
dCount := dCount + 1;
if v not = 0 and dCount = 3 then begin
r( cPos // 1 ) := ",";
cPos   := cPos - 1;
dCount := 0
end if_v_ne_0_and_dCount_eq_3
end for_cPos;
r( cPos // 1 ) := if n < 0 then "-" else " ";
writeon( s_w := 0, r )
end writeonWithCommas ;
begin % task test cases                                                  %
integer prevI, prevR, root, p7;
write( "Integer square roots of 0..65 (values the same as the previous one not shown):" );
write();
prevR := prevI := -1;
for i := 0 until 65 do begin
root := iSqrt( i );
if root not = prevR then begin
prevR := root;
prevI := i;
writeon( i_w := 1, s_w := 0, " ", i, ":", root )
end
else if prevI = i - 1 then writeon( "..." );
end for_i ;
write();
% integer square roots of odd powers of 7                            %
write( "Integer square roots of 7^n, odd n" );
write( " n|           7^n|    isqrt(7^n)" );
write( " -+--------------+--------------" );
p7 := 7;
for p := 1 step 2 until 9 do begin
write( i_w := 2, s_w := 0, p );
writeon( s_w := 0, "|" ); writeonWithCommas(        p7   );
writeon( s_w := 0, "|" ); writeonWithCommas( iSqrt( p7 ) );
p7 := p7 * 49
end for_p
end.```
Output:
```Integer square roots of 0..65 (values the same as the previous one not shown):
0:0 1:1... 4:2... 9:3... 16:4... 25:5... 36:6... 49:7... 64:8...

Integer square roots of 7^n, odd n
n|           7^n|    isqrt(7^n)
-+--------------+--------------
1|             7|             2
3|           343|            18
5|        16,807|           129
7|       823,543|           907
9|    40,353,607|         6,352
```

## ALGOL-M

The code presented here follows the task description. But be warned: there is a bug lurking in the algorithm as presented. The statement q := q * 4 in the first while loop will overflow the limits of ALGOL-M's integer data type (-16,383 to +16,383) for any value of x greater than 4095 and trigger an endless loop. The output has been put into columnar form to avoid what would otherwise be an ugly mess on a typical 80 column display.

```BEGIN

COMMENT
RETURN INTEGER SQUARE ROOT OF N USING QUADRATIC RESIDUE
ALGORITHM. WARNING: THE FUNCTION WILL FAIL FOR X GREATER
THAN 4095;
INTEGER FUNCTION ISQRT(X);
INTEGER X;
BEGIN
INTEGER Q, R, Z, T;
Q := 1;
WHILE Q <= X DO
Q := Q * 4;   % WARNING! OVERFLOW YIELDS 0 %
Z := X;
R := 0;
WHILE Q > 1 DO
BEGIN
Q := Q / 4;
T := Z - R - Q;
R := R / 2;
IF T >= 0 THEN
BEGIN
Z := T;
R := R + Q;
END;
END;
ISQRT := R;
END;

COMMENT - LET'S EXERCISE THE FUNCTION;

INTEGER I, COL;
WRITE("INTEGER SQUARE ROOT OF FIRST 65 NUMBERS:");
WRITE("");
COL := 1;
FOR I := 1 STEP 1 UNTIL 65 DO
BEGIN
WRITEON(ISQRT(I));
COL := COL + 1;
IF COL > 10 THEN
BEGIN
WRITE("");
COL := 1;
END;
END;

WRITE("");
WRITE("     N    7^N  ISQRT");
WRITE("--------------------");
COMMENT - ODD POWERS OF 7 GREATER THAN 3 WILL CAUSE OVERFLOW;
FOR I := 1 STEP 2 UNTIL 3 DO
BEGIN
INTEGER POW7;
POW7 := 7**I;
WRITE(I, POW7, ISQRT(POW7));
END;
WRITE("THAT'S ALL. GOODBYE.");

END```

An alternative to the quadratic residue approach will allow calculation of the integer square root for the full range of signed integer values supported by ALGOL-M. (The output is identical.)

```% RETURN INTEGER SQUARE ROOT OF N %
INTEGER FUNCTION ISQRT(N);
INTEGER N;
BEGIN
INTEGER R1, R2;
R1 := N;
R2 := 1;
WHILE R1 > R2 DO
BEGIN
R1 := (R1+R2) / 2;
R2 := N / R1;
END;
ISQRT := R1;
END;```
Output:
```INTEGER SQUARE ROOT OF FIRST 65 NUMBERS:
1     1     1     2     2     2     2     2     3     3
3     3     3     3     3     4     4     4     4     4
4     4     4     4     5     5     5     5     5     5
5     5     5     5     5     6     6     6     6     6
6     6     6     6     6     6     6     6     7     7
7     7     7     7     7     7     7     7     7     7
7     7     7     8     8

N    7^N  ISQRT
--------------------
1     7     2
3   343    18
THAT'S ALL. GOODBYE.
```

## AppleScript

The odd-powers-of-7 part of the task is limited by the precision of AppleScript reals.

```on isqrt(x)
set q to 1
repeat until (q > x)
set q to q * 4
end repeat
set z to x
set r to 0
repeat while (q > 1)
set q to q div 4
set t to z - r - q
set r to r div 2
if (t > -1) then
set z to t
set r to r + q
end if
end repeat

return r
end isqrt

on intToText(n, separator)
set output to ""
repeat until (n < 1000)
set output to separator & (text 2 thru 4 of ((1000 + (n mod 1000) as integer) as text)) & output
set n to n div 1000
end repeat

return (n as integer as text) & output
end intToText

-- Get the integer and power results.
set {integerResults, powerResults} to {{}, {}}
repeat with x from 0 to 65
set end of integerResults to isqrt(x)
end repeat
repeat with p from 1 to 73 by 2
set x to 7 ^ p
if (x > 1.0E+15) then exit repeat -- Beyond the precision of AppleScript reals.
set end of powerResults to "7^" & p & tab & "(" & intToText(x, ",") & "):" & (tab & tab & intToText(isqrt(x), ","))
end repeat
-- Format and output.
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to space
set output to {"Isqrts of integers from 0 to 65:", space & integerResults, ¬
"Isqrts of odd powers of 7 from 1 to " & (p - 2) & ":", powerResults}
set AppleScript's text item delimiters to linefeed
set output to output as text
set AppleScript's text item delimiters to astid

return output

```
Output:
```"Isqrts of integers from 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Isqrts of odd powers of 7 from 1 to 17:
7^1	(7):		2
7^3	(343):		18
7^5	(16,807):		129
7^7	(823,543):		907
7^9	(40,353,607):		6,352
7^11	(1,977,326,743):		44,467
7^13	(96,889,010,407):		311,269
7^15	(4,747,561,509,943):		2,178,889
7^17	(232,630,513,987,207):		15,252,229"
```

## APL

Works in Dyalog APL

``` i←{x←⍵
q←(×∘4)⍣{⍺>x}⊢1
⊃{  r z q←⍵
q←⌊q÷4
t←(z-r)-q
r←⌊r÷2
z←z t[1+t≥0]
r←r+q×t≥0
r z q
}⍣{ r z q←⍺
q≤1
}⊢0 x q
}
```
Output:
```      i¨⍳65
1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6

6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

(⎕fr⎕pp)←1287 34
↑{⍵ (7*⍵) (i 7*⍵)}¨1,1+2×⍳10
1                  7         2
3                343        18
5              16807       129
7             823543       907
9           40353607      6352
11         1977326743     44467
13        96889010407    311269
15      4747561509943   2178889
17    232630513987207  15252229
19  11398895185373143 106765608
21 558545864083284007 747359260
```

## Arturo

```commatize: function [x][
reverse join.with:"," map split.every: 3 split reverse to :string x => join
]

isqrt: function [x][
num: new x
q: new 1
r: new 0

while [q =< num]-> shl.safe 'q 2
while [q > 1][
shr 'q 2
t: (num-r)-q
shr 'r 1
if t >= 0 [
num: t
r: new r+q
]
]
return r
]

print map 0..65 => isqrt

loop range 1 .step: 2 72 'n ->
print [n "\t" commatize isqrt 7^n]
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
1 	 2
3 	 18
5 	 129
7 	 907
9 	 6,352
11 	 44,467
13 	 311,269
15 	 2,178,889
17 	 15,252,229
19 	 106,765,608
21 	 747,359,260
23 	 5,231,514,822
25 	 36,620,603,758
27 	 256,344,226,312
29 	 1,794,409,584,184
31 	 12,560,867,089,291
33 	 87,926,069,625,040
35 	 615,482,487,375,282
37 	 4,308,377,411,626,977
39 	 30,158,641,881,388,842
41 	 211,110,493,169,721,897
43 	 1,477,773,452,188,053,281
45 	 10,344,414,165,316,372,973
47 	 72,410,899,157,214,610,812
49 	 506,876,294,100,502,275,687
51 	 3,548,134,058,703,515,929,815
53 	 24,836,938,410,924,611,508,707
55 	 173,858,568,876,472,280,560,953
57 	 1,217,009,982,135,305,963,926,677
59 	 8,519,069,874,947,141,747,486,745
61 	 59,633,489,124,629,992,232,407,216
63 	 417,434,423,872,409,945,626,850,517
65 	 2,922,040,967,106,869,619,387,953,625
67 	 20,454,286,769,748,087,335,715,675,381
69 	 143,180,007,388,236,611,350,009,727,669
71 	 1,002,260,051,717,656,279,450,068,093,686```

## ATS

Big integers are achieved via the GNU Multiple Precision interface, which enforces proper memory management, without the need for a garbage collector. I felt that, in ATS, using GMP would be almost as simple as writing isqrt only for machine-native integers. One has to try it, to see the advantage of using a compiler that tells you when you have left out an initialization or a "free".

One might note the construction of a "comma'd numeral" by consing of a linked list. This method will work in any Lisp, ML, etc., also. Here the lists are of type "list_vt" and so enforce proper memory management, even in absence of a garbage collector.

```(*

Compile with "myatscc isqrt.dats", thus obtaining an executable called
"isqrt".

##myatsccdef=\
patscc -O2 \
-I"\${PATSHOME}/contrib/atscntrb" \
-IATS "\${PATSHOME}/contrib/atscntrb" \
-D_GNU_SOURCE -DATS_MEMALLOC_LIBC \
-o \$fname(\$1) \$1 -lgmp

*)

(* An interface to GNU Multiple Precision. The type system will help
ensure that you do "mpz_clear" on whatever you allocate. *)

(* As of this writing, gmp.dats is empty, but it does no harm to

fn
find_a_power_of_4_greater_than_x
(x : &mpz,            (* Input. *)
q : &mpz? >> mpz)    (* Output. *)
: void =
let
fun
loop (x : &mpz, q : &mpz) : void =
if 0 <= mpz_cmp (x, q) then
begin
mpz_mul (q, 4u);
loop (x, q)
end
in
mpz_init_set (q, 1u);
loop (x, q)
end

fn
isqrt_and_remainder
(x : &mpz,            (* Input. *)
r : &mpz? >> mpz,    (* Output: square root. *)
z : &mpz? >> mpz)    (* Output: remainder. *)
: void =
let
fun
loop (q : &mpz, z : &mpz, r : &mpz, t : &mpz) : void =
if 0 < mpz_cmp (q, 1u) then
begin
mpz_tdiv_q (q, 4u);
mpz_set_mpz (t, z);
mpz_sub (t, r);
mpz_sub (t, q);
mpz_tdiv_q (r, 2u);
if 0 <= mpz_cmp (t, 0u) then
begin
mpz_set_mpz (z, t);
end;
loop (q, z, r, t);
end

var q : mpz
var t : mpz
in
find_a_power_of_4_greater_than_x (x, q);
mpz_init_set (z, x);
mpz_init_set (r, 0u);
mpz_init (t);

loop (q, z, r, t);

mpz_clear (q);
mpz_clear (t);
end

fn
isqrt (x : &mpz,                (* Input. *)
r : &mpz? >> mpz)        (* Output: square root. *)
: void =
let
var z : mpz
in
isqrt_and_remainder (x, r, z);
mpz_clear (z);
end

fn
print_n_spaces (n : uint) : void =
let
var i : [i : nat] uint i
in
for (i := 0u; i < n; i := succ i)
print! (" ")
end

fn
print_with_commas (n           : &mpz,
num_columns : uint) : void =
let
fun
make_list (q   : &mpz,
r   : &mpz,
lst : List0_vt char,
i   : uint) : List_vt char =
if mpz_cmp (q, 0u) = 0 then
lst
else
let
val _ = mpz_tdiv_qr (q, r, 10u)
val ones_place = mpz_get_int (r)
val digit = int2char0 (ones_place + char2i '0')
in
if i = 3u then
let
val lst = list_vt_cons (',', lst)
val lst = list_vt_cons (digit, lst)
in
make_list (q, r, lst, 1u)
end
else
let
val lst = list_vt_cons (digit, lst)
in
make_list (q, r, lst, succ i)
end
end

var q : mpz
var r : mpz

val _ = mpz_init_set (q, n)
val _ = mpz_init (r)
val char_lst = make_list (q, r, list_vt_nil (), 0u)
val _ = mpz_clear (q)
val _ = mpz_clear (r)

fun
print_and_consume_lst (char_lst : List0_vt char) : void =
case+ char_lst of
| ~ list_vt_nil () => ()
| ~ list_vt_cons (head, tail) =>
begin
print_and_consume_lst (tail);
end

prval _ = lemma_list_vt_param (char_lst)
val len = i2u (list_vt_length (char_lst))
in
assertloc (len <= num_columns);
print_n_spaces (num_columns - len);
print_and_consume_lst (char_lst)
end

fn
do_the_roots_of_0_to_65 () : void =
let
var i : mpz
in
mpz_init_set (i, 0u);
while (mpz_cmp (i, 65u) <= 0)
let
var r : mpz
in
isqrt (i, r);
fprint (stdout_ref, r);
print! (" ");
mpz_clear (r);
end;
mpz_clear (i);
end

fn
do_the_roots_of_odd_powers_of_7 () : void =
let
var seven : mpz
var seven_raised_i : mpz
var i_mpz : mpz
var i : [i : pos] uint i
in
mpz_init_set (seven, 7u);
mpz_init (seven_raised_i);
mpz_init (i_mpz);
for (i := 1u; i <= 73u; i := succ (succ i))
let
var r : mpz
in
mpz_pow_uint (seven_raised_i, seven, i);
isqrt (seven_raised_i, r);
mpz_set_uint (i_mpz, i);
print_with_commas (i_mpz, 2u);
print! (" ");
print_with_commas (seven_raised_i, 84u);
print! (" ");
print_with_commas (r, 43u);
print! ("\n");
mpz_clear (r);
end;
mpz_clear (seven);
mpz_clear (seven_raised_i);
mpz_clear (i_mpz);
end

implement
main0 () =
begin
print! ("isqrt(i) for 0 <= i <= 65:\n\n");
do_the_roots_of_0_to_65 ();
print! ("\n\n\n");
print! ("isqrt(7**i) for 1 <= i <= 73, i odd:\n\n");
print! (" i                                                                                 7**i                                  sqrt(7**i)\n");
print! ("-----------------------------------------------------------------------------------------------------------------------------------\n");
do_the_roots_of_odd_powers_of_7 ();
end```
Output:
```\$ myatscc isqrt.dats
\$ ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## BASIC256

```print "Integer square root of first 65 numbers:"
for n = 1 to 65
print ljust(isqrt(n),3);
next n
print : print
print "Integer square root of odd powers of 7"
print "  n                 7^n       isqrt"
print "-"*36
for n = 1 to 21 step 2
pow7 = int(7 ^ n)
print rjust(n,3);rjust(pow7,20);rjust(isqrt(pow7),12)
next n
end

function isqrt(x)
q = 1
while q <= x
q *= 4
end while
r = 0
while q > 1
q /= 4
t = x - r - q
r /= 2
if t >= 0 then
x = t
r += q
end if
end while
return int(r)
end function```

## C

Translation of: C++

Up to 64-bit limits with no big int library.

```#include <stdint.h>
#include <stdio.h>

int64_t isqrt(int64_t x) {
int64_t q = 1, r = 0;
while (q <= x) {
q <<= 2;
}
while (q > 1) {
int64_t t;
q >>= 2;
t = x - r - q;
r >>= 1;
if (t >= 0) {
x = t;
r += q;
}
}
return r;
}

int main() {
int64_t p;
int n;

printf("Integer square root for numbers 0 to 65:\n");
for (n = 0; n <= 65; n++) {
printf("%lld ", isqrt(n));
}
printf("\n\n");

printf("Integer square roots of odd powers of 7 from 1 to 21:\n");
printf(" n |              7 ^ n | isqrt(7 ^ n)\n");
p = 7;
for (n = 1; n <= 21; n += 2, p *= 49) {
printf("%2d | %18lld | %12lld\n", n, p, isqrt(p));
}
}
```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 21:
n |              7 ^ n | isqrt(7 ^ n)
1 |                  7 |            2
3 |                343 |           18
5 |              16807 |          129
7 |             823543 |          907
9 |           40353607 |         6352
11 |         1977326743 |        44467
13 |        96889010407 |       311269
15 |      4747561509943 |      2178889
17 |    232630513987207 |     15252229
19 |  11398895185373143 |    106765608
21 | 558545864083284007 |    747359260```

## C++

Library: Boost
```#include <iomanip>
#include <iostream>
#include <sstream>
#include <boost/multiprecision/cpp_int.hpp>

using big_int = boost::multiprecision::cpp_int;

template <typename integer>
integer isqrt(integer x) {
integer q = 1;
while (q <= x)
q <<= 2;
integer r = 0;
while (q > 1) {
q >>= 2;
integer t = x - r - q;
r >>= 1;
if (t >= 0) {
x = t;
r += q;
}
}
return r;
}

std::string commatize(const big_int& n) {
std::ostringstream out;
out << n;
std::string str(out.str());
std::string result;
size_t digits = str.size();
result.reserve(4 * digits/3);
for (size_t i = 0; i < digits; ++i) {
if (i > 0 && i % 3 == digits % 3)
result += ',';
result += str[i];
}
return result;
}

int main() {
std::cout << "Integer square root for numbers 0 to 65:\n";
for (int n = 0; n <= 65; ++n)
std::cout << isqrt(n) << ' ';
std::cout << "\n\n";

std::cout << "Integer square roots of odd powers of 7 from 1 to 73:\n";
const int power_width = 83, isqrt_width = 42;
std::cout << " n |"
<< std::setw(power_width) << "7 ^ n" << " |"
<< std::setw(isqrt_width) << "isqrt(7 ^ n)"
<< '\n';
std::cout << std::string(6 + power_width + isqrt_width, '-') << '\n';
big_int p = 7;
for (int n = 1; n <= 73; n += 2, p *= 49) {
std::cout << std::setw(2) << n << " |"
<< std::setw(power_width) << commatize(p) << " |"
<< std::setw(isqrt_width) << commatize(isqrt(p))
<< '\n';
}
return 0;
}
```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 73:
n |                                                                              7 ^ n |                              isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 |                                                                                  7 |                                         2
3 |                                                                                343 |                                        18
5 |                                                                             16,807 |                                       129
7 |                                                                            823,543 |                                       907
9 |                                                                         40,353,607 |                                     6,352
11 |                                                                      1,977,326,743 |                                    44,467
13 |                                                                     96,889,010,407 |                                   311,269
15 |                                                                  4,747,561,509,943 |                                 2,178,889
17 |                                                                232,630,513,987,207 |                                15,252,229
19 |                                                             11,398,895,185,373,143 |                               106,765,608
21 |                                                            558,545,864,083,284,007 |                               747,359,260
23 |                                                         27,368,747,340,080,916,343 |                             5,231,514,822
25 |                                                      1,341,068,619,663,964,900,807 |                            36,620,603,758
27 |                                                     65,712,362,363,534,280,139,543 |                           256,344,226,312
29 |                                                  3,219,905,755,813,179,726,837,607 |                         1,794,409,584,184
31 |                                                157,775,382,034,845,806,615,042,743 |                        12,560,867,089,291
33 |                                              7,730,993,719,707,444,524,137,094,407 |                        87,926,069,625,040
35 |                                            378,818,692,265,664,781,682,717,625,943 |                       615,482,487,375,282
37 |                                         18,562,115,921,017,574,302,453,163,671,207 |                     4,308,377,411,626,977
39 |                                        909,543,680,129,861,140,820,205,019,889,143 |                    30,158,641,881,388,842
41 |                                     44,567,640,326,363,195,900,190,045,974,568,007 |                   211,110,493,169,721,897
43 |                                  2,183,814,375,991,796,599,109,312,252,753,832,343 |                 1,477,773,452,188,053,281
45 |                                107,006,904,423,598,033,356,356,300,384,937,784,807 |                10,344,414,165,316,372,973
47 |                              5,243,338,316,756,303,634,461,458,718,861,951,455,543 |                72,410,899,157,214,610,812
49 |                            256,923,577,521,058,878,088,611,477,224,235,621,321,607 |               506,876,294,100,502,275,687
51 |                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 |             3,548,134,058,703,515,929,815
53 |                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 |            24,836,938,410,924,611,508,707
55 |                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 |           173,858,568,876,472,280,560,953
57 |                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 |         1,217,009,982,135,305,963,926,677
59 |                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 |         8,519,069,874,947,141,747,486,745
61 |              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 |        59,633,489,124,629,992,232,407,216
63 |            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 |       417,434,423,872,409,945,626,850,517
65 |          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 |     2,922,040,967,106,869,619,387,953,625
67 |        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 |    20,454,286,769,748,087,335,715,675,381
69 |     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 |   143,180,007,388,236,611,350,009,727,669
71 |  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
```

## C#

```using System;
using static System.Console;
using BI = System.Numerics.BigInteger;

class Program {

static BI isqrt(BI x) { BI q = 1, r = 0, t; while (q <= x) q <<= 2; while (q > 1) {
q >>= 2; t = x - r - q; r >>= 1; if (t >= 0) { x = t; r += q; } } return r; }

static void Main() { const int max = 73, smax = 65;
int power_width = ((BI.Pow(7, max).ToString().Length / 3) << 2) + 3,
isqrt_width = (power_width + 1) >> 1;
WriteLine("Integer square root for numbers 0 to {0}:", smax);
for (int n = 0; n <= smax; ++n) Write("{0} ",
(n / 10).ToString().Replace("0", " ")); WriteLine();
for (int n = 0; n <= smax; ++n) Write("{0} ", n % 10); WriteLine();
WriteLine(new String('-', (smax << 1) + 1));
for (int n = 0; n <= smax; ++n) Write("{0} ", isqrt(n));
WriteLine("\n\nInteger square roots of odd powers of 7 from 1 to {0}:", max);
string s = string.Format("[0,2] |[1,{0}:n0] |[2,{1}:n0]",
power_width, isqrt_width).Replace("[", "{").Replace("]", "}");
WriteLine(s, "n", "7 ^ n", "isqrt(7 ^ n)");
WriteLine(new String('-', power_width + isqrt_width + 6));
BI p = 7; for (int n = 1; n <= max; n += 2, p *= 49)
WriteLine (s, n, p, isqrt(p)); }
}
```
Output:
```Integer square root for numbers 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
-----------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 73:
n |                                                                              7 ^ n |                              isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 |                                                                                  7 |                                         2
3 |                                                                                343 |                                        18
5 |                                                                             16,807 |                                       129
7 |                                                                            823,543 |                                       907
9 |                                                                         40,353,607 |                                     6,352
11 |                                                                      1,977,326,743 |                                    44,467
13 |                                                                     96,889,010,407 |                                   311,269
15 |                                                                  4,747,561,509,943 |                                 2,178,889
17 |                                                                232,630,513,987,207 |                                15,252,229
19 |                                                             11,398,895,185,373,143 |                               106,765,608
21 |                                                            558,545,864,083,284,007 |                               747,359,260
23 |                                                         27,368,747,340,080,916,343 |                             5,231,514,822
25 |                                                      1,341,068,619,663,964,900,807 |                            36,620,603,758
27 |                                                     65,712,362,363,534,280,139,543 |                           256,344,226,312
29 |                                                  3,219,905,755,813,179,726,837,607 |                         1,794,409,584,184
31 |                                                157,775,382,034,845,806,615,042,743 |                        12,560,867,089,291
33 |                                              7,730,993,719,707,444,524,137,094,407 |                        87,926,069,625,040
35 |                                            378,818,692,265,664,781,682,717,625,943 |                       615,482,487,375,282
37 |                                         18,562,115,921,017,574,302,453,163,671,207 |                     4,308,377,411,626,977
39 |                                        909,543,680,129,861,140,820,205,019,889,143 |                    30,158,641,881,388,842
41 |                                     44,567,640,326,363,195,900,190,045,974,568,007 |                   211,110,493,169,721,897
43 |                                  2,183,814,375,991,796,599,109,312,252,753,832,343 |                 1,477,773,452,188,053,281
45 |                                107,006,904,423,598,033,356,356,300,384,937,784,807 |                10,344,414,165,316,372,973
47 |                              5,243,338,316,756,303,634,461,458,718,861,951,455,543 |                72,410,899,157,214,610,812
49 |                            256,923,577,521,058,878,088,611,477,224,235,621,321,607 |               506,876,294,100,502,275,687
51 |                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 |             3,548,134,058,703,515,929,815
53 |                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 |            24,836,938,410,924,611,508,707
55 |                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 |           173,858,568,876,472,280,560,953
57 |                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 |         1,217,009,982,135,305,963,926,677
59 |                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 |         8,519,069,874,947,141,747,486,745
61 |              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 |        59,633,489,124,629,992,232,407,216
63 |            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 |       417,434,423,872,409,945,626,850,517
65 |          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 |     2,922,040,967,106,869,619,387,953,625
67 |        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 |    20,454,286,769,748,087,335,715,675,381
69 |     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 |   143,180,007,388,236,611,350,009,727,669
71 |  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
```

## CLU

```% This program uses the 'bigint' cluster from PCLU's 'misc.lib'

% Integer square root of a bigint
isqrt = proc (x: bigint) returns (bigint)
% Initialize a couple of bigints we will reuse
own zero: bigint := bigint\$i2bi(0)
own one: bigint := bigint\$i2bi(1)
own two: bigint := bigint\$i2bi(2)
own four: bigint := bigint\$i2bi(4)

q: bigint := one
while q <= x do q := q * four end

t: bigint
z: bigint := x
r: bigint := zero
while q>one do
q := q / four
t := z - r - q
r := r / two
if t >= zero then
z := t
r := r + q
end
end
return(r)
end isqrt

% Format a bigint using commas
fmt = proc (x: bigint) returns (string)
own zero: bigint := bigint\$i2bi(0)
own ten: bigint := bigint\$i2bi(10)

if x=zero then return("0") end
out: array[char] := array[char]\$[]
ds: int := 0
while x>zero do
array[char]\$addl(out, char\$i2c(bigint\$bi2i(x // ten) + 48))
x := x / ten
ds := ds + 1
if x~=zero cand ds//3=0 then
end
end
return(string\$ac2s(out))
end fmt

start_up = proc ()
po: stream := stream\$primary_output()

% print square roots from 0..65
stream\$putl(po, "isqrt of 0..65:")
for i: int in int\$from_to(0, 65) do
stream\$puts(po, fmt(isqrt(bigint\$i2bi(i))) || " ")
end

% print square roots of odd powers
stream\$putl(po, "\n\nisqrt of odd powers of 7:")
seven: bigint := bigint\$i2bi(7)
for p: int in int\$from_to_by(1, 73, 2) do
stream\$puts(po, "isqrt(7^")
stream\$putright(po, int\$unparse(p), 2)
stream\$puts(po, ") = ")
stream\$putright(po, fmt(isqrt(seven ** bigint\$i2bi(p))), 41)
stream\$putl(po, "")
end
end start_up```
Output:
```isqrt of 0..65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt of odd powers of 7:
isqrt(7^ 1) =                                         2
isqrt(7^ 3) =                                        18
isqrt(7^ 5) =                                       129
isqrt(7^ 7) =                                       907
isqrt(7^ 9) =                                     6,352
isqrt(7^11) =                                    44,467
isqrt(7^13) =                                   311,269
isqrt(7^15) =                                 2,178,889
isqrt(7^17) =                                15,252,229
isqrt(7^19) =                               106,765,608
isqrt(7^21) =                               747,359,260
isqrt(7^23) =                             5,231,514,822
isqrt(7^25) =                            36,620,603,758
isqrt(7^27) =                           256,344,226,312
isqrt(7^29) =                         1,794,409,584,184
isqrt(7^31) =                        12,560,867,089,291
isqrt(7^33) =                        87,926,069,625,040
isqrt(7^35) =                       615,482,487,375,282
isqrt(7^37) =                     4,308,377,411,626,977
isqrt(7^39) =                    30,158,641,881,388,842
isqrt(7^41) =                   211,110,493,169,721,897
isqrt(7^43) =                 1,477,773,452,188,053,281
isqrt(7^45) =                10,344,414,165,316,372,973
isqrt(7^47) =                72,410,899,157,214,610,812
isqrt(7^49) =               506,876,294,100,502,275,687
isqrt(7^51) =             3,548,134,058,703,515,929,815
isqrt(7^53) =            24,836,938,410,924,611,508,707
isqrt(7^55) =           173,858,568,876,472,280,560,953
isqrt(7^57) =         1,217,009,982,135,305,963,926,677
isqrt(7^59) =         8,519,069,874,947,141,747,486,745
isqrt(7^61) =        59,633,489,124,629,992,232,407,216
isqrt(7^63) =       417,434,423,872,409,945,626,850,517
isqrt(7^65) =     2,922,040,967,106,869,619,387,953,625
isqrt(7^67) =    20,454,286,769,748,087,335,715,675,381
isqrt(7^69) =   143,180,007,388,236,611,350,009,727,669
isqrt(7^71) = 1,002,260,051,717,656,279,450,068,093,686
isqrt(7^73) = 7,015,820,362,023,593,956,150,476,655,802```

## COBOL

The COBOL compiler used here is limited to 18-digit math, meaning 7^19 is the largest odd power of 7 that can be calculated.

```       IDENTIFICATION DIVISION.
PROGRAM-ID. I-SQRT.

DATA DIVISION.
WORKING-STORAGE SECTION.
03 X          PIC 9(18).
03 Q          PIC 9(18).
03 Z          PIC 9(18).
03 T          PIC S9(18).
03 R          PIC 9(18).

01 TO-65-VARS.
03 ISQRT-N    PIC 99.
03 DISP-LN    PIC X(22) VALUE SPACES.
03 DISP-FMT   PIC Z9.
03 PTR        PIC 99 VALUE 1.

01 BIG-SQRT-VARS.
03 POW-7      PIC 9(18) VALUE 7.
03 POW-N      PIC 99 VALUE 1.
03 POW-N-OUT  PIC Z9.
03 POW-7-OUT  PIC Z(10).

PROCEDURE DIVISION.
BEGIN.
PERFORM SQRTS-TO-65.
PERFORM BIG-SQRTS.
STOP RUN.

SQRTS-TO-65.
PERFORM DISP-SMALL-SQRT
VARYING ISQRT-N FROM 0 BY 1
UNTIL ISQRT-N IS GREATER THAN 65.

DISP-SMALL-SQRT.
MOVE ISQRT-N TO X.
PERFORM ISQRT.
MOVE R TO DISP-FMT.
STRING DISP-FMT DELIMITED BY SIZE INTO DISP-LN
WITH POINTER PTR.
IF PTR IS GREATER THAN 22,
DISPLAY DISP-LN,
MOVE 1 TO PTR.

BIG-SQRTS.
PERFORM BIG-SQRT 10 TIMES.

BIG-SQRT.
MOVE POW-7 TO X.
PERFORM ISQRT.
MOVE POW-N TO POW-N-OUT.
MOVE R TO POW-7-OUT.
DISPLAY "ISQRT(7^" POW-N-OUT ") = " POW-7-OUT.
MULTIPLY 49 BY POW-7.

ISQRT.
MOVE 1 TO Q.
PERFORM MUL-Q-BY-4 UNTIL Q IS GREATER THAN X.
MOVE X TO Z.
MOVE ZERO TO R.
PERFORM ISQRT-STEP UNTIL Q IS NOT GREATER THAN 1.

MUL-Q-BY-4.
MULTIPLY 4 BY Q.

ISQRT-STEP.
DIVIDE 4 INTO Q.
COMPUTE T = Z - R - Q.
DIVIDE 2 INTO R.
IF T IS NOT LESS THAN ZERO,
MOVE T TO Z,
```
Output:
``` 0 1 1 1 2 2 2 2 2 3 3
3 3 3 3 3 4 4 4 4 4 4
4 4 4 5 5 5 5 5 5 5 5
5 5 5 6 6 6 6 6 6 6 6
6 6 6 6 6 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 8 8
ISQRT(7^ 1) =          2
ISQRT(7^ 3) =         18
ISQRT(7^ 5) =        129
ISQRT(7^ 7) =        907
ISQRT(7^ 9) =       6352
ISQRT(7^11) =      44467
ISQRT(7^13) =     311269
ISQRT(7^15) =    2178889
ISQRT(7^17) =   15252229
ISQRT(7^19) =  106765608```

## Common Lisp

Translation of: Scheme

The program is wrapped in a Roswell script. On a POSIX system with Roswell installed, you can simply run the script.

The code is an "imperative" translation of the "functional" Scheme.

Side notes:

• Straight translations from Scheme to Common Lisp run the risk of running properly with some CL compilers but not others, due to the prevalence of tail calls in Scheme. Common Lisp does not require proper tail calls, although CL compilers do optimize such calls, to varying degrees depending on the compiler.
• The simple program here would not require tail call optimization at all; the depth of recursion would be too small. I felt it better to write the CL in "imperative" style nonetheless.
• Not all Scheme compilers make all tail calls proper, either, at least by default. This, however, is nonstandard behavior.

```#!/bin/sh
#|-*- mode:lisp -*-|#
#|
exec ros -Q -- \$0 "\$@"
|#
(progn ;;init forms
(ros:ensure-asdf)
)

(defpackage :ros.script.isqrt.3860764029
(:use :cl))
(in-package :ros.script.isqrt.3860764029)

;;
;; The Rosetta Code integer square root task, in Common Lisp.
;;
;; I translate the tail recursions of the Scheme as regular loops in
;; Common Lisp, although CL compilers most often can optimize tail
;; recursions of the kind. They are not required to, however.
;;
;; As a result, the CL is actually closer to the task's pseudocode
;; than is the Scheme.
;;
;; (The Scheme, by the way, could have been written much as follows,
;; using "set!" where the CL has "setf", and with other such
;; "linguistic" changes.)
;;

(defun find-a-power-of-4-greater-than-x (x)
(let ((q 1))
(loop until (< x q)
do (setf q (* 4 q)))
q))

(defun isqrt+remainder (x)
(let ((q (find-a-power-of-4-greater-than-x x))
(z x)
(r 0))
(loop until (= q 1)
do (progn (setf q (/ q 4))
(let ((z1 (- z r q)))
(setf r (/ r 2))
(when (<= 0 z1)
(setf z z1)
(setf r (+ r q))))))
(values r z)))

(defun rosetta_code_isqrt (x)
(nth-value 0 (isqrt+remainder x)))

(defun main (&rest argv)
(declare (ignorable argv))
(format t "isqrt(i) for ~D <= i <= ~D:~2%" 0 65)
(loop for i from 0 to 64
do (format t "~D " (isqrt i)))
(format t "~D~3%" (isqrt 65))
(format t "isqrt(7**i) for ~D <= i <= ~D, i odd:~2%" 1 73)
(format t "~2@A ~84@A ~43@A~%" "i" "7**i" "sqrt(7**i)")
(format t "~A~%" (make-string 131 :initial-element #\-))
(loop for i from 1 to 73 by 2
for 7**i = (expt 7 i)
for root = (rosetta_code_isqrt 7**i)
do (format t "~2D ~84:D ~43:D~%" i 7**i root)))

;;; vim: set ft=lisp lisp:
```
Output:
```\$ sh isqrt.ros
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## Cowgol

```include "cowgol.coh";

# Integer square root
sub isqrt(x: uint32): (x0: uint32) is
x0 := x >> 1;
if x0 == 0 then
x0 := x;
return;
end if;
loop
var x1 := (x0 + x/x0) >> 1;
if x1 >= x0 then
break;
end if;
x0 := x1;
end loop;
end sub;

# Power
sub pow(x: uint32, n: uint8): (r: uint32) is
r := 1;
while n > 0 loop
r := r * x;
n := n - 1;
end loop;
end sub;

# Print integer square roots of 0..65
var n: uint32 := 0;
var col: uint8 := 11;
while n <= 65 loop
print_i32(isqrt(n));
col := col - 1;
if col == 0 then
print_nl();
col := 11;
else
print_char(' ');
end if;
n := n + 1;
end loop;

# Cowgol only supports 32-bit integers out of the box, so only powers of 7
# up to 7^11 are printed
var x: uint8 := 0;
while x <= 11 loop
print("isqrt(7^");
print_i8(x);
print(") = ");
print_i32(isqrt(pow(7, x)));
print_nl();
x := x + 1;
end loop;```
Output:
```0 1 1 1 2 2 2 2 2 3 3
3 3 3 3 3 4 4 4 4 4 4
4 4 4 5 5 5 5 5 5 5 5
5 5 5 6 6 6 6 6 6 6 6
6 6 6 6 6 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 8 8
isqrt(7^0) = 1
isqrt(7^1) = 2
isqrt(7^2) = 7
isqrt(7^3) = 18
isqrt(7^4) = 49
isqrt(7^5) = 129
isqrt(7^6) = 343
isqrt(7^7) = 907
isqrt(7^8) = 2401
isqrt(7^9) = 6352
isqrt(7^10) = 16807
isqrt(7^11) = 44467```

## Craft Basic

```alert "integer square root of first 65 numbers:"

for n = 1 to 65

let x = n
gosub isqrt
print r

next n

alert "integer square root of odd powers of 7"
cls
cursor 1, 1

for n = 1 to 21 step 2

let x = 7 ^ n
gosub isqrt
print "isqrt of 7 ^ ", n, " = ", r

next n

end

sub isqrt

let q = 1

do

if q <= x then

let q = q * 4

endif

wait

loop q <= x

let r = 0

do

if q > 1 then

let q = q / 4
let t = x - r - q
let r = r / 2

if t >= 0 then

let x = t
let r = r + q

endif

endif

loop q > 1

let r = int(r)

return
```
Output:
```integer square root of first 65 numbers:
1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
integer square root of odd powers of 7:
isqrt of 7 ^ 1 = 2
isqrt of 7 ^ 3 = 18
isqrt of 7 ^ 5 = 129
isqrt of 7 ^ 7 = 907

isqrt of 7 ^ 9 = 6352```

## D

Translation of: Kotlin
```import std.bigint;
import std.conv;
import std.exception;
import std.range;
import std.regex;
import std.stdio;

auto commatize(in char[] txt, in uint start=0, in uint step=3, in string ins=",") @safe
in {
assert(step > 0);
} body {
if (start > txt.length || step > txt.length)    {
return txt;
}

// First number may begin with digit or decimal point. Exponents ignored.
enum decFloField = ctRegex!("[0-9]*\\.[0-9]+|[0-9]+");

auto matchDec = matchFirst(txt[start .. \$], decFloField);
if (!matchDec) {
return txt;
}

// Within a decimal float field:
// A decimal integer field to commatize is positive and not after a point.
enum decIntField = ctRegex!("(?<=\\.)|[1-9][0-9]*");
// A decimal fractional field is preceded by a point, and is only digits.
enum decFracField = ctRegex!("(?<=\\.)[0-9]+");

return txt[0 .. start] ~ matchDec.pre ~ matchDec.hit
.replace!(m => m.hit.retro.chunks(step).join(ins).retro)(decIntField)
.replace!(m => m.hit.chunks(step).join(ins))(decFracField)
~ matchDec.post;
}

auto commatize(BigInt v) {
return commatize(v.to!string);
}

BigInt sqrt(BigInt x) {
enforce(x >= 0);

auto q = BigInt(1);
while (q <= x) {
q <<= 2;
}
auto z = x;
auto r = BigInt(0);
while (q > 1) {
q >>= 2;
auto t = z;
t -= r;
t -= q;
r >>= 1;
if (t >= 0) {
z = t;
r += q;
}
}
return r;
}

void main() {
writeln("The integer square root of integers from 0 to 65 are:");
foreach (i; 0..66) {
write(sqrt(BigInt(i)), ' ');
}
writeln;

writeln("The integer square roots of powers of 7 from 7^1 up to 7^73 are:");
writeln("power                                    7 ^ power                                                 integer square root");
writeln("----- --------------------------------------------------------------------------------- -----------------------------------------");
auto pow7 = BigInt(7);
immutable bi49 = BigInt(49);
for (int i = 1; i <= 73; i += 2) {
writefln("%2d %84s %41s", i, pow7.commatize, sqrt(pow7).commatize);
pow7 *= bi49;
}
}
```
Output:
```The integer square root of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of powers of 7 from 7^1 up to 7^73 are:
power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802```

See #Pascal.

## Draco

Because all the intermediate values have to fit in a signed 32-bit integer, the largest power of 7 for which the square root can be calculated is 7^10.

```/* Integer square root using quadratic residue method */
proc nonrec isqrt(ulong x) ulong:
ulong q, z, r;
long t;

q := 1;
while q <= x do q := q << 2 od;

z := x;
r := 0;
while q > 1 do
q := q >> 2;
t := z - r - q;
r := r >> 1;
if t >= 0 then
z := t;
r := r + q
fi
od;

r
corp

proc nonrec main() void:
byte x;
ulong pow7;

/* print isqrt(0) ... isqrt(65) */
for x from 0 upto 65 do
write(isqrt(x):2);
if x % 11 = 10 then writeln() fi
od;

/* print isqrt(7^0) thru isqrt(7^10) */
pow7 := 1;
for x from 0 upto 10 do
writeln("isqrt(7^", x:2, ") = ", isqrt(pow7):5);
pow7 := pow7 * 7
od
corp```
Output:
``` 0 1 1 1 2 2 2 2 2 3 3
3 3 3 3 3 4 4 4 4 4 4
4 4 4 5 5 5 5 5 5 5 5
5 5 5 6 6 6 6 6 6 6 6
6 6 6 6 6 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 8 8
isqrt(7^ 0) =     1
isqrt(7^ 1) =     2
isqrt(7^ 2) =     7
isqrt(7^ 3) =    18
isqrt(7^ 4) =    49
isqrt(7^ 5) =   129
isqrt(7^ 6) =   343
isqrt(7^ 7) =   907
isqrt(7^ 8) =  2401
isqrt(7^ 9) =  6352
isqrt(7^10) = 16807```

## EasyLang

Translation of: Lua
```func isqrt x .
q = 1
while q <= x
q *= 4
.
while q > 1
q = q div 4
t = x - r - q
r = r div 2
if t >= 0
x = t
r = r + q
.
.
return r
.
print "Integer square roots from 0 to 65:"
for n = 0 to 65
write isqrt n & " "
.
print ""
print ""
print "Integer square roots of 7^n"
p = 7
n = 1
while n <= 21
print n & " " & isqrt p
n = n + 2
p = p * 49
.```

## F#

```// Find Integer Floor sqrt of a Large Integer. Nigel Galloway: July 17th., 2020
let Isqrt n=let rec fN i g l=match(l>0I,i-g-l) with
(true,e) when e>=0I->fN e (g/2I+l) (l/4I)
|(true,_)           ->fN i (g/2I)   (l/4I)
|_                  ->g
fN n 0I (let rec fG g=if g>n then g/4I else fG (g*4I) in fG 1I)
[0I..65I]|>Seq.iter(Isqrt>>string>>printf "%s "); printfn "\n"
let fN n=7I**n in [1..2..73]|>Seq.iter(fN>>Isqrt>>printfn "%a" (fun n g -> n.Write("{0:#,#}", g)))
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

2
18
129
907
6,352
44,467
311,269
2,178,889
15,252,229
106,765,608
747,359,260
5,231,514,822
36,620,603,758
256,344,226,312
1,794,409,584,184
12,560,867,089,291
87,926,069,625,040
615,482,487,375,282
4,308,377,411,626,977
30,158,641,881,388,842
211,110,493,169,721,897
1,477,773,452,188,053,281
10,344,414,165,316,372,973
72,410,899,157,214,610,812
506,876,294,100,502,275,687
3,548,134,058,703,515,929,815
24,836,938,410,924,611,508,707
173,858,568,876,472,280,560,953
1,217,009,982,135,305,963,926,677
8,519,069,874,947,141,747,486,745
59,633,489,124,629,992,232,407,216
417,434,423,872,409,945,626,850,517
2,922,040,967,106,869,619,387,953,625
20,454,286,769,748,087,335,715,675,381
143,180,007,388,236,611,350,009,727,669
1,002,260,051,717,656,279,450,068,093,686
7,015,820,362,023,593,956,150,476,655,802
```

## Factor

The `isqrt` word is a straightforward translation of the pseudocode from the task description using lexical variables.

Works with: Factor version 0.99 2020-07-03
```USING: formatting io kernel locals math math.functions
math.ranges prettyprint sequences tools.memory.private ;

:: isqrt ( x -- n )
1 :> q!
[ q x > ] [ q 4 * q! ] until
x 0 :> ( z! r! )
[ q 1 > ] [
q 4 /i q!
z r - q - :> t
r -1 shift r!
t 0 >= [
t z!
r q + r!
] when
] while
r ;

"Integer square root for numbers 0 to 65 (inclusive):" print
66 <iota> [ bl ] [ isqrt pprint ] interleave nl nl

: align ( str str str -- ) "%2s%85s%44s\n" printf ;
: show ( n -- ) dup 7 swap ^ dup isqrt [ commas ] tri@ align ;

"x" "7^x" "isqrt(7^x)" align
"-" "---" "----------" align
1 73 2 <range> [ show ] each
```
Output:
```Integer square root for numbers 0 to 65 (inclusive):
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

x                                                                                  7^x                                  isqrt(7^x)
-                                                                                  ---                                  ----------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802
```

## Fish

A function (isolated stack; remove the 1[ and ] instructions to create "inline" version) that calculates and outputs the square root of the top of the stack/input number.

```1[:>:r:@@:@,\;
]~\$\!?={:,2+/n
```

## Forth

Only handles odd powers of 7 up to 7^21.

```: d., ( n -- ) \ write double precision int, commatized.
tuck dabs
<# begin  2dup 1.000 d>  while  # # # [char] , hold  repeat #s rot sign #>
type space ;

: .,  ( n -- ) \ write integer commatized.
s>d d., ;

: 4*  s" 2 lshift" evaluate ; immediate
: 4/  s" 2 rshift" evaluate ; immediate

: isqrt-mod ( n -- z r )  \ n = r^2 + z
1 begin 2dup >= while 4* repeat
0 locals| r q z |
begin q 1 > while
q 4/ to q
z r - q - \ t
r 2/ to r
dup 0>= if
to z
r q + to r
else
drop
then
repeat z r ;

: isqrt  isqrt-mod nip ;

." Integer square roots from 0 to 65:" cr
66 0 do  i isqrt .  loop cr ;

." Integer square roots of 7^n" cr
7 11 0 do
i 2* 1+ 2 .r 3 spaces
dup isqrt ., cr
49 *
loop ;

```

This version of the core word does not require locals.

```: sqrt-rem                             ( n -- sqrt rem)
>r 0 1 begin dup r@ > 0= while 4 * repeat
begin                                \ find a power of 4 greater than TORS
dup 1 >                            \ compute while greater than unity
while
2/ 2/ swap over over + negate r@ + \ integer divide by 4
dup 0< if drop 2/ else r> drop >r 2/ over + then swap
repeat drop r>                       ( sqrt rem)
;

: isqrt-mod sqrt-rem swap ;
```
Output:
```Integer square roots from 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of 7^n
1   2
3   18
5   129
7   907
9   6,352
11   44,467
13   311,269
15   2,178,889
17   15,252,229
19   106,765,608
21   747,359,260
```

## Fortran

```MODULE INTEGER_SQUARE_ROOT
IMPLICIT NONE

CONTAINS

! Convert string representation number to string with comma digit separation
FUNCTION COMMATIZE(NUM) RESULT(OUT_STR)
INTEGER(16), INTENT(IN) :: NUM
INTEGER(16) I
CHARACTER(83) :: TEMP, OUT_STR

WRITE(TEMP, '(I0)') NUM

OUT_STR = ""

DO I=0, LEN_TRIM(TEMP)-1
IF (MOD(I, 3) .EQ. 0 .AND. I .GT. 0 .AND. I .LT. LEN_TRIM(TEMP)) THEN
OUT_STR = "," // TRIM(OUT_STR)
END IF
OUT_STR = TEMP(LEN_TRIM(TEMP)-I:LEN_TRIM(TEMP)-I) // TRIM(OUT_STR)
END DO
END FUNCTION COMMATIZE

! Calculate the integer square root for a given integer
FUNCTION ISQRT(NUM)
INTEGER(16), INTENT(IN) :: NUM
INTEGER(16) :: ISQRT
INTEGER(16) :: Q, Z, R, T

Q = 1
Z = NUM
R = 0
T = 0

DO WHILE (Q .LE. NUM)
Q = Q * 4
END DO

DO WHILE (Q .GT. 1)
Q = Q / 4
T = Z - R - Q
R = R / 2

IF (T .GE. 0) THEN
Z = T
R = R + Q
END IF
END DO

ISQRT = R
END FUNCTION ISQRT

END MODULE INTEGER_SQUARE_ROOT

! Demonstration of integer square root for numbers 0-65 followed by odd powers of 7
! from 1-73. Currently this demo takes significant time for numbers above 43
PROGRAM ISQRT_DEMO
USE INTEGER_SQUARE_ROOT
IMPLICIT NONE

INTEGER(16), PARAMETER :: MIN_NUM_HZ = 0
INTEGER(16), PARAMETER :: MAX_NUM_HZ = 65
INTEGER(16), PARAMETER :: POWER_BASE = 7
INTEGER(16), PARAMETER :: POWER_MIN = 1
INTEGER(16), PARAMETER :: POWER_MAX = 73
INTEGER(16), DIMENSION(MAX_NUM_HZ-MIN_NUM_HZ+1) :: VALUES

INTEGER(16) :: I

WRITE(*,'(A, I0, A, I0)') "Integer square root for numbers ", MIN_NUM_HZ, " to ", MAX_NUM_HZ

DO I=1, SIZE(VALUES)
VALUES(I) = ISQRT(MIN_NUM_HZ+I)
END DO

WRITE(*,'(100I2)') VALUES
WRITE(*,*) NEW_LINE('A')

WRITE(*,*) REPEAT("-", 8+83*2)

DO I=POWER_MIN,POWER_MAX, 2
WRITE(*,'(I2, A, A, A, A)') I, " | " // COMMATIZE(7**I), " | ", COMMATIZE(ISQRT(7**I))
END DO

END PROGRAM ISQRT_DEMO
```
```Integer square root for numbers 0 to 65
0 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

n | 7^n                                                                                 | isqrt(7^n)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1 | 7                                                                                   | 2
3 | 343                                                                                 | 18
5 | 16,807                                                                              | 129
7 | 823,543                                                                             | 907
9 | 40,353,607                                                                          | 6,352
11 | 1,977,326,743                                                                       | 44,467
13 | 96,889,010,407                                                                      | 311,269
15 | 4,747,561,509,943                                                                   | 2,178,889
17 | 232,630,513,987,207                                                                 | 15,252,229
19 | 11,398,895,185,373,143                                                              | 106,765,608
21 | 558,545,864,083,284,007                                                             | 747,359,260
23 | 27,368,747,340,080,916,343                                                          | 5,231,514,822
25 | 1,341,068,619,663,964,900,807                                                       | 36,620,603,758
27 | 65,712,362,363,534,280,139,543                                                      | 256,344,226,312
29 | 3,219,905,755,813,179,726,837,607                                                   | 1,794,409,584,184
31 | 157,775,382,034,845,806,615,042,743                                                 | 12,560,867,089,291
33 | 7,730,993,719,707,444,524,137,094,407                                               | 87,926,069,625,040
35 | 378,818,692,265,664,781,682,717,625,943                                             | 615,482,487,375,282
37 | 18,562,115,921,017,574,302,453,163,671,207                                          | 4,308,377,411,626,977
39 | 909,543,680,129,861,140,820,205,019,889,143                                         | 30,158,641,881,388,842
41 | 44,567,640,326,363,195,900,190,045,974,568,007                                      | 211,110,493,169,721,897
43 | 2,183,814,375,991,796,599,109,312,252,753,832,343                                   | 1,477,773,452,188,053,281
```

## FreeBASIC

Odd powers up to 7^21 are shown; more would require an arbitrary precision library that would just add bloat without being illustrative.

```function isqrt( byval x as ulongint ) as ulongint
dim as ulongint q = 1, r
dim as longint t
while q <= x
q = q shl 2
wend
while q > 1
q = q shr 2
t = x - r - q
r = r shr 1
if t >= 0  then
x  = t
r += q
end if
wend
return r
end function

function commatize( byval N as string ) as string
dim as string bloat = ""
dim as uinteger c = 0
while N<>""
bloat = right(N,1) + bloat
N = left(N, len(N)-1)
c += 1
if c mod 3 = 0 and N<>"" then bloat = "," + bloat
wend
return bloat
end function

for i as ulongint = 0 to 65
print isqrt(i);" ";
next i
print

dim as string ns
for i as uinteger = 1 to 22 step 2
ns = str(isqrt(7^i))
print i, commatize(ns)
next i```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
1             2
3             18
5             129
7             907
9             6,352
11            44,467
13            311,269
15            2,178,889
17            15,252,229
19            106,765,608
21            747,359,260```

## FutureBasic

```include "NSLog.incl"

local fn IntSqrt( x as SInt64 ) as SInt64
SInt64 q = 1, z = x, r = 0, t
do
q = q * 4
until ( q > x )
while( q > 1 )
q = q / 4 : t = z - r - q : r = r / 2
if ( t > -1 ) then  z = t : r = r + q
wend
end fn = r

SInt64      p
NSInteger   n
CFNumberRef tempNum
CFStringRef tempStr

NSLog( @"Integer square root for numbers 0 to 65:" )

for n = 0 to 65
NSLog( @"%lld \b", fn IntSqrt( n ) )
next
NSLog( @"\n" )

NSLog( @"Integer square roots of odd powers of 7 from 1 to 21:" )
NSLog( @" n |              7 ^ n | fn IntSqrt(7 ^ n)" )
p = 7
for n = 1 to 21 step 2
tempNum = fn NumberWithLongLong( fn IntSqrt(p) )
tempStr = fn NumberDescriptionWithLocale( tempNum, fn LocaleCurrent )
NSLog( @"%2d | %18lld | %12s", n, p, fn StringUTF8String( tempStr ) )
p = p * 49
next

HandleEvents```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 21:
n |              7 ^ n | fn IntSqrt(7 ^ n)
1 |                  7 |            2
3 |                343 |           18
5 |              16807 |          129
7 |             823543 |          907
9 |           40353607 |        6,352
11 |         1977326743 |       44,467
13 |        96889010407 |      311,269
15 |      4747561509943 |    2,178,889
17 |    232630513987207 |   15,252,229
19 |  11398895185373143 |  106,765,608
21 | 558545864083284007 |  747,359,260
```

## Go

Go's big.Int type already has a built-in integer square root function but, as the point of this task appears to be to compute it using a particular algorithm, we re-code it from the pseudo-code given in the task description.

```package main

import (
"fmt"
"log"
"math/big"
)

var zero = big.NewInt(0)
var one = big.NewInt(1)

func isqrt(x *big.Int) *big.Int {
if x.Cmp(zero) < 0 {
log.Fatal("Argument cannot be negative.")
}
q := big.NewInt(1)
for q.Cmp(x) <= 0 {
q.Lsh(q, 2)
}
z := new(big.Int).Set(x)
r := big.NewInt(0)
for q.Cmp(one) > 0 {
q.Rsh(q, 2)
t := new(big.Int)
t.Sub(t, r)
t.Sub(t, q)
r.Rsh(r, 1)
if t.Cmp(zero) >= 0 {
z.Set(t)
}
}
return r
}

func commatize(s string) string {
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}

func main() {
fmt.Println("The integer square roots of integers from 0 to 65 are:")
for i := int64(0); i <= 65; i++ {
fmt.Printf("%d ", isqrt(big.NewInt(i)))
}
fmt.Println()
fmt.Println("\nThe integer square roots of powers of 7 from 7^1 up to 7^73 are:\n")
fmt.Println("power                                    7 ^ power                                                 integer square root")
fmt.Println("----- --------------------------------------------------------------------------------- -----------------------------------------")
pow7 := big.NewInt(7)
bi49 := big.NewInt(49)
for i := 1; i <= 73; i += 2 {
fmt.Printf("%2d %84s %41s\n", i, commatize(pow7.String()), commatize(isqrt(pow7).String()))
pow7.Mul(pow7, bi49)
}
}
```
Output:
```The integer square roots of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

The integer square roots of powers of 7 from 7^1 up to 7^73 are:

power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
```

```import Data.Bits

isqrt :: Integer -> Integer
isqrt n = go n 0 (q `shiftR` 2)
where
q = head \$ dropWhile (< n) \$ iterate (`shiftL` 2) 1
go z r 0 = r
go z r q = let t = z - r - q
in if t >= 0
then go t (r `shiftR` 1 + q) (q `shiftR` 2)
else go z (r `shiftR` 1) (q `shiftR` 2)

main = do
print \$ isqrt <\$> [1..65]
mapM_ print \$ zip [1,3..73] (isqrt <\$> iterate (49 *) 7)
```
```*Main> main
[0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8]
(1,2)
(3,18)
(5,129)
(7,907)
(9,6352)
(11,44467)
(13,311269)
(15,2178889)
(17,15252229)
(19,106765608)
(21,747359260)
(23,5231514822)
(25,36620603758)
(27,256344226312)
(29,1794409584184)
(31,12560867089291)
(33,87926069625040)
(35,615482487375282)
(37,4308377411626977)
(39,30158641881388842)
(41,211110493169721897)
(43,1477773452188053281)
(45,10344414165316372973)
(47,72410899157214610812)
(49,506876294100502275687)
(51,3548134058703515929815)
(53,24836938410924611508707)
(55,173858568876472280560953)
(57,1217009982135305963926677)
(59,8519069874947141747486745)
(61,59633489124629992232407216)
(63,417434423872409945626850517)
(65,2922040967106869619387953625)
(67,20454286769748087335715675381)
(69,143180007388236611350009727669)
(71,1002260051717656279450068093686)
(73,7015820362023593956150476655802)```

## Icon

```link numbers                    # For the "commas" procedure.

procedure main ()
write ("isqrt(i) for 0 <= i <= 65:")
write ()
roots_of_0_to_65()
write ()
write ()
write ("isqrt(7**i) for 1 <= i <= 73, i odd:")
write ()
printf ("%2s %84s %43s\n", "i", "7**i", "sqrt(7**i)")
write (repl("-", 131))
roots_of_odd_powers_of_7()
end

procedure roots_of_0_to_65 ()
local i

every i := 0 to 64 do writes (isqrt(i), " ")
write (isqrt(65))
end

procedure roots_of_odd_powers_of_7 ()
local i, power_of_7, root

every i := 1 to 73 by 2 do {
power_of_7 := 7^i
root := isqrt(power_of_7)
printf ("%2d %84s %43s\n", i, commas(power_of_7), commas(root))
}
end

procedure isqrt (x)
local q, z, r, t

q := find_a_power_of_4_greater_than_x (x)
z := x
r := 0
while 1 < q do {
q := ishift(q, -2)
t := z - r - q
r := ishift(r, -1)
if 0 <= t then {
z := t
r +:= q
}
}
return r
end

procedure find_a_power_of_4_greater_than_x (x)
local q

q := 1
while q <= x do q := ishift(q, 2)
return q
end
```
Output:
```\$ icont -s -u -o isqrt isqrt-in-Icon.icn && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## J

Three implementations given. The floating point method is best for small square roots, Newton's method is fastest for extended integers. isqrt adapted from the page preamble. Note that the "floating point method" is exact when arbitrary precision integers or rational numbers are used.

```isqrt_float=: <.@:%:
isqrt_newton=: 9&\$: :(x:@:<.@:-:@:(] + x:@:<.@:%)^:_&>~&:x:)

align=: (|.~ i.&' ')"1
comma=: (' ' -.~ [: }: [: , [: (|.) _3 (',' ,~ |.)\ |.)@":&>
While=: {{ u^:(0-.@:-:v)^:_ }}

isqrt=: 3 :0&>
y =. x: y
NB. q is a power of 4 that's greater than y.  Append 0 0 under binary representation
q =. y (,&0 0x&.:#:@:])While>: 1x
z =. y               NB. set  z  to the value of y.
r =. 0x              NB. initialize  r  to zero.
while. 1 < q do.     NB. perform while  q > unity.
q =. _2&}.&.:#: q   NB. integer divide by 4 (-2 drop under binary representation)
t =. (z - r) - q    NB. compute value of  t.
r =. }:&.:#: r      NB. integer divide by  two. (curtail under binary representation)
if. 0 <: t do.
z =. t             NB. set  z  to value of t
r =. r + q         NB. compute new value of r
end.
end.
NB. r  is now the  isqrt(y). (most recent value computed)
NB. Sidenote: Also, Z is now the remainder after square root
NB. ie. r^2 + z = y, so if z = 0 then x is a perfect square
NB. r , z
)
```

The first line here shows that the simplest approach (the "floating point square root") is treated specially by J.

```   <.@%: 1000000000000000000000000000000000000000000000000x
1000000000000000000000000

(,. isqrt_float) 7x ^ 20 21x
79792266297612001 282475249
558545864083284007 747359260

(,. isqrt_newton) 7x ^ 20 21x
79792266297612001 282475249
558545864083284007 747359260

align comma (,. isqrt) 7 ^&x: 1 2 p. i. 37
7
2

343
18

16,807
129

823,543
907

40,353,607
6,352

1,977,326,743
44,467

96,889,010,407
311,269

4,747,561,509,943
2,178,889

232,630,513,987,207
15,252,229

11,398,895,185,373,143
106,765,608

558,545,864,083,284,007
747,359,260

27,368,747,340,080,916,343
5,231,514,822

1,341,068,619,663,964,900,807
36,620,603,758

65,712,362,363,534,280,139,543
256,344,226,312

3,219,905,755,813,179,726,837,607
1,794,409,584,184

157,775,382,034,845,806,615,042,743
12,560,867,089,291

7,730,993,719,707,444,524,137,094,407
87,926,069,625,040

378,818,692,265,664,781,682,717,625,943
615,482,487,375,282

18,562,115,921,017,574,302,453,163,671,207
4,308,377,411,626,977

909,543,680,129,861,140,820,205,019,889,143
30,158,641,881,388,842

44,567,640,326,363,195,900,190,045,974,568,007
211,110,493,169,721,897

2,183,814,375,991,796,599,109,312,252,753,832,343
1,477,773,452,188,053,281

107,006,904,423,598,033,356,356,300,384,937,784,807
10,344,414,165,316,372,973

5,243,338,316,756,303,634,461,458,718,861,951,455,543
72,410,899,157,214,610,812

256,923,577,521,058,878,088,611,477,224,235,621,321,607
506,876,294,100,502,275,687

12,589,255,298,531,885,026,341,962,383,987,545,444,758,743
3,548,134,058,703,515,929,815

616,873,509,628,062,366,290,756,156,815,389,726,793,178,407
24,836,938,410,924,611,508,707

30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943
173,858,568,876,472,280,560,953

1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207
1,217,009,982,135,305,963,926,677

72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143
8,519,069,874,947,141,747,486,745

3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007
59,633,489,124,629,992,232,407,216

174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343
417,434,423,872,409,945,626,850,517

8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807
2,922,040,967,106,869,619,387,953,625

418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543
20,454,286,769,748,087,335,715,675,381

20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607
143,180,007,388,236,611,350,009,727,669

1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743
1,002,260,051,717,656,279,450,068,093,686

49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407
7,015,820,362,023,593,956,150,476,655,802

NB. isqrt_float is exact for large integers
align comma (,.isqrt_float),7^73x
49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407
7,015,820,362,023,593,956,150,476,655,802

NB. Newton's method result matches isqrt
(isqrt_newton -: isqrt)7 ^&x: 1 2 p. i. 37
1

NB. An order of magnitude faster and one tenth the space, in j

timespacex 'isqrt_newton 7 ^&x: 1 2 p. i. 37'
0.038085 39552
timespacex 'isqrt 7 ^&x: 1 2 p. i. 37'
0.367744 319712

NB. but not as fast as isqrt_float, nor as space efficient
timespacex 'isqrt_float 7 ^&x: 1 2 p. i. 37'
0.0005145 152192```

Note that isqrt_float (`<.@%:`) is, mechanically, a different operation from floating point square root followed by floor. This difference is probably best thought of as a type issue. For example, `<.%:7^73x` (without the `@`) produces a result which has the comma delimited integer representation of 7,015,820,362,023,594,877,495,225,090 rather than the 7,015,820,362,023,593,956,150,476,655,802 which we would get from integer square root.

## Java

Translation of: Kotlin
```import java.math.BigInteger;

public class Isqrt {
private static BigInteger isqrt(BigInteger x) {
if (x.compareTo(BigInteger.ZERO) < 0) {
throw new IllegalArgumentException("Argument cannot be negative");
}
var q = BigInteger.ONE;
while (q.compareTo(x) <= 0) {
q = q.shiftLeft(2);
}
var z = x;
var r = BigInteger.ZERO;
while (q.compareTo(BigInteger.ONE) > 0) {
q = q.shiftRight(2);
var t = z;
t = t.subtract(r);
t = t.subtract(q);
r = r.shiftRight(1);
if (t.compareTo(BigInteger.ZERO) >= 0) {
z = t;
}
}
return r;
}

public static void main(String[] args) {
System.out.println("The integer square root of integers from 0 to 65 are:");
for (int i = 0; i <= 65; i++) {
System.out.printf("%s ", isqrt(BigInteger.valueOf(i)));
}
System.out.println();

System.out.println("The integer square roots of powers of 7 from 7^1 up to 7^73 are:");
System.out.println("power                                    7 ^ power                                                 integer square root");
System.out.println("----- --------------------------------------------------------------------------------- -----------------------------------------");
var pow7 = BigInteger.valueOf(7);
var bi49 = BigInteger.valueOf(49);
for (int i = 1; i < 74; i += 2) {
System.out.printf("%2d %,84d %,41d\n", i, pow7, isqrt(pow7));
pow7 = pow7.multiply(bi49);
}
}
}
```
Output:
```The integer square root of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of powers of 7 from 7^1 up to 7^73 are:
power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802```

## jq

Translation of: Julia

The following program takes advantage of the support for unbounded-precision

integer arithmetic provided by gojq, the Go implementation of jq, but it can also be run, with different numerical results, using the C implementation.
```# For gojq
def idivide(\$j):
. as \$i
| (\$i % \$j) as \$mod
| (\$i - \$mod) / \$j ;

# input should be non-negative
def isqrt:
. as \$x
| 1 | until(. > \$x; . * 4) as \$q
| {\$q, \$x, r: 0}
| until( .q <= 1;
.q |= idivide(4)
| .t = .x - .r - .q
| .r |= idivide(2)
| if .t >= 0
then .x = .t
| .r += .q
else . end).r ;

def power(\$n):
. as \$in
| reduce range(0;\$n) as \$i (1; . * \$in);

def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] + .;

"The integer square roots of integers from 0 to 65 are:",
[range(0;66) | isqrt],
"",
"The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:",
("power" + " "*16 + "7 ^ power" + " "*70 + "integer square root"),

(range( 1;74;2) as \$i
| (7 | power(\$i)) as \$p
Output:

Invocation: gojq -ncr -f isqrt.jq

```The integer square roots of integers from 0 to 65 are:
[0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8]

The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:
power                7 ^ power                                                                      integer square root
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                                16807                                         129
7                                                                               823543                                         907
9                                                                             40353607                                        6352
11                                                                           1977326743                                       44467
13                                                                          96889010407                                      311269
15                                                                        4747561509943                                     2178889
17                                                                      232630513987207                                    15252229
19                                                                    11398895185373143                                   106765608
21                                                                   558545864083284007                                   747359260
23                                                                 27368747340080916343                                  5231514822
25                                                               1341068619663964900807                                 36620603758
27                                                              65712362363534280139543                                256344226312
29                                                            3219905755813179726837607                               1794409584184
31                                                          157775382034845806615042743                              12560867089291
33                                                         7730993719707444524137094407                              87926069625040
35                                                       378818692265664781682717625943                             615482487375282
37                                                     18562115921017574302453163671207                            4308377411626977
39                                                    909543680129861140820205019889143                           30158641881388842
41                                                  44567640326363195900190045974568007                          211110493169721897
43                                                2183814375991796599109312252753832343                         1477773452188053281
45                                              107006904423598033356356300384937784807                        10344414165316372973
47                                             5243338316756303634461458718861951455543                        72410899157214610812
49                                           256923577521058878088611477224235621321607                       506876294100502275687
51                                         12589255298531885026341962383987545444758743                      3548134058703515929815
53                                        616873509628062366290756156815389726793178407                     24836938410924611508707
55                                      30226801971775055948247051683954096612865741943                    173858568876472280560953
57                                    1481113296616977741464105532513750734030421355207                   1217009982135305963926677
59                                   72574551534231909331741171093173785967490646405143                   8519069874947141747486745
61                                 3556153025177363557255317383565515512407041673852007                  59633489124629992232407216
63                               174251498233690814305510551794710260107945042018748343                 417434423872409945626850517
65                              8538323413450849900970017037940802745289307058918668807                2922040967106869619387953625
67                            418377847259091645147530834859099334519176045887014771543               20454286769748087335715675381
69                          20500514515695490612229010908095867391439626248463723805607              143180007388236611350009727669
71                        1004525211269079039999221534496697502180541686174722466474743             1002260051717656279450068093686
73                       49221735352184872959961855190338177606846542622561400857262407             7015820362023593956150476655802
```

## Julia

Translation of: Go

Julia also has a built in isqrt() function which works on integer types, but the function integer_sqrt is shown for the task.

```using Formatting

function integer_sqrt(x)
@assert(x >= 0)
q = one(x)
while q <= x
q <<= 2
end
z, r = x, zero(x)
while q > 1
q >>= 2
t = z - r - q
r >>= 1
if t >= 0
z = t
r += q
end
end
return r
end

println("The integer square roots of integers from 0 to 65 are:")
println(integer_sqrt.(collect(0:65)))

println("\nThe integer square roots of odd powers of 7 from 7^1 up to 7^73 are:\n")
println("power", " "^36, "7 ^ power", " "^60, "integer square root")
println("----- ", "-"^80, " ------------------------------------------")
pow7 = big"7"
for i in 1:2:73
end
```
Output:
```The integer square roots of integers from 0 to 65 are:
[0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8]

The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:

power                                    7 ^ power                                                            integer square root
----- -------------------------------------------------------------------------------- ------------------------------------------
1                                                                                   7                                          2
3                                                                                 343                                         18
5                                                                              16,807                                        129
7                                                                             823,543                                        907
9                                                                          40,353,607                                      6,352
11                                                                       1,977,326,743                                     44,467
13                                                                      96,889,010,407                                    311,269
15                                                                   4,747,561,509,943                                  2,178,889
17                                                                 232,630,513,987,207                                 15,252,229
19                                                              11,398,895,185,373,143                                106,765,608
21                                                             558,545,864,083,284,007                                747,359,260
23                                                          27,368,747,340,080,916,343                              5,231,514,822
25                                                       1,341,068,619,663,964,900,807                             36,620,603,758
27                                                      65,712,362,363,534,280,139,543                            256,344,226,312
29                                                   3,219,905,755,813,179,726,837,607                          1,794,409,584,184
31                                                 157,775,382,034,845,806,615,042,743                         12,560,867,089,291
33                                               7,730,993,719,707,444,524,137,094,407                         87,926,069,625,040
35                                             378,818,692,265,664,781,682,717,625,943                        615,482,487,375,282
37                                          18,562,115,921,017,574,302,453,163,671,207                      4,308,377,411,626,977
39                                         909,543,680,129,861,140,820,205,019,889,143                     30,158,641,881,388,842
41                                      44,567,640,326,363,195,900,190,045,974,568,007                    211,110,493,169,721,897
43                                   2,183,814,375,991,796,599,109,312,252,753,832,343                  1,477,773,452,188,053,281
45                                 107,006,904,423,598,033,356,356,300,384,937,784,807                 10,344,414,165,316,372,973
47                               5,243,338,316,756,303,634,461,458,718,861,951,455,543                 72,410,899,157,214,610,812
49                             256,923,577,521,058,878,088,611,477,224,235,621,321,607                506,876,294,100,502,275,687
51                          12,589,255,298,531,885,026,341,962,383,987,545,444,758,743              3,548,134,058,703,515,929,815
53                         616,873,509,628,062,366,290,756,156,815,389,726,793,178,407             24,836,938,410,924,611,508,707
55                      30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943            173,858,568,876,472,280,560,953
57                   1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207          1,217,009,982,135,305,963,926,677
59                  72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143          8,519,069,874,947,141,747,486,745
61               3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007         59,633,489,124,629,992,232,407,216
63             174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343        417,434,423,872,409,945,626,850,517
65           8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807      2,922,040,967,106,869,619,387,953,625
67         418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543     20,454,286,769,748,087,335,715,675,381
69      20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607    143,180,007,388,236,611,350,009,727,669
71   1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743  1,002,260,051,717,656,279,450,068,093,686
73  49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407  7,015,820,362,023,593,956,150,476,655,802
```

## Kotlin

Translation of: Go
```import java.math.BigInteger

fun isqrt(x: BigInteger): BigInteger {
if (x < BigInteger.ZERO) {
throw IllegalArgumentException("Argument cannot be negative")
}
var q = BigInteger.ONE
while (q <= x) {
q = q.shiftLeft(2)
}
var z = x
var r = BigInteger.ZERO
while (q > BigInteger.ONE) {
q = q.shiftRight(2)
var t = z
t -= r
t -= q
r = r.shiftRight(1)
if (t >= BigInteger.ZERO) {
z = t
r += q
}
}
return r
}

fun main() {
println("The integer square root of integers from 0 to 65 are:")
for (i in 0..65) {
print("\${isqrt(BigInteger.valueOf(i.toLong()))} ")
}
println()

println("The integer square roots of powers of 7 from 7^1 up to 7^73 are:")
println("power                                    7 ^ power                                                 integer square root")
println("----- --------------------------------------------------------------------------------- -----------------------------------------")
var pow7 = BigInteger.valueOf(7)
val bi49 = BigInteger.valueOf(49)
for (i in (1..73).step(2)) {
println("%2d %,84d %,41d".format(i, pow7, isqrt(pow7)))
pow7 *= bi49
}
}
```
Output:
```The integer square root of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of powers of 7 from 7^1 up to 7^73 are:
power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802```

## Lua

Translation of: C
```function isqrt(x)
local q = 1
local r = 0
while q <= x do
q = q << 2
end
while q > 1 do
q = q >> 2
local t = x - r - q
r = r >> 1
if t >= 0 then
x = t
r = r + q
end
end
return r
end

print("Integer square root for numbers 0 to 65:")
for n=0,65 do
io.write(isqrt(n) .. ' ')
end
print()
print()

print("Integer square roots of oddd powers of 7 from 1 to 21:")
print(" n |              7 ^ n | isqrt(7 ^ n)")
local p = 7
local n = 1
while n <= 21 do
print(string.format("%2d | %18d | %12d", n, p, isqrt(p)))
----------------------
n = n + 2
p = p * 49
end
```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7
7 7 8 8

Integer square roots of oddd powers of 7 from 1 to 21:
n |              7 ^ n | isqrt(7 ^ n)
3 |                343 |           18
5 |              16807 |          129
7 |             823543 |          907
9 |           40353607 |         6352
11 |         1977326743 |        44467
13 |        96889010407 |       311269
15 |      4747561509943 |      2178889
17 |    232630513987207 |     15252229
19 |  11398895185373143 |    106765608
21 | 558545864083284007 |    747359260```

## M2000 Interpreter

Using various types up to 7^35

```module integer_square_root (f=-2) {
function IntSqrt(x as long long) {
long long q=1, z=x, t, r
do q*=4&& : until (q>x)
while q>1&&
q|div 4&&:t=z-r-q:r|div 2&&
if t>-1&& then z=t:r+= q
end while
=r
}
long i
print #f, "The integer square root of integers from 0 to 65 are:"
for i=0 to 65
print #f, IntSqrt(i)+" ";
next
print #f
print #f, "Using Long Long Type"
print #f, "The integer square roots of powers of 7 from 7^1 up to 7^21 are:"
for i=1 to 21 step 2 {
print #f, "IntSqrt(7^"+i+")="+(IntSqrt(7&&^i))+" of 7^"+i+" ("+(7&&^I)+")"
}
print #f
function IntSqrt(x as decimal) {
decimal q=1, z=x, t, r
do q*=4 : until (q>x)
while q>1
q/=4:t=z-r-q:r/=2
if t>-1 then z=t:r+= q
end while
=r
}
print #f, "Using Decimal Type"
print #f, "The integer square roots of powers of 7 from 7^23 up to 7^33 are:"
decimal j,p
for i=23 to 33 step 2 {
p=1:for j=1 to i:p*=7@:next
print #f, "IntSqrt(7^"+i+")="+(IntSqrt(p))+" of 7^"+i+" ("+p+")"
}
print #f

function IntSqrt(x as double) {
double q=1, z=x, t, r
do q*=4 : until (q>x)
while q>1
q/=4:t=z-r-q:r/=2
if t>-1 then z=t:r+= q
end while
=r
}
print #f, "Using Double Type"
print #f, "The integer square roots of powers of 7 from 7^19 up to 7^35 are:"
for i=19 to 35 step 2 {
print #f, "IntSqrt(7^"+i+")="+(IntSqrt(7^i))+" of 7^"+i+" ("+(7^i)+")"
}
print #f
}
open "" for output as #f  // f = -2 now, direct output to screen
integer_square_root
close #f
open "out.txt" for output as #f
integer_square_root f
close #f
Output:
```The integer square root of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Using Long Long Type
The integer square roots of powers of 7 from 7^1 up to 7^21 are:
IntSqrt(7^1)=2 of 7^1 (7)
IntSqrt(7^3)=18 of 7^3 (343)
IntSqrt(7^5)=129 of 7^5 (16807)
IntSqrt(7^7)=907 of 7^7 (823543)
IntSqrt(7^9)=6352 of 7^9 (40353607)
IntSqrt(7^11)=44467 of 7^11 (1977326743)
IntSqrt(7^13)=311269 of 7^13 (96889010407)
IntSqrt(7^15)=2178889 of 7^15 (4747561509943)
IntSqrt(7^17)=15252229 of 7^17 (232630513987207)
IntSqrt(7^19)=106765608 of 7^19 (11398895185373143)
IntSqrt(7^21)=747359260 of 7^21 (558545864083284007)

Using Decimal Type
The integer square roots of powers of 7 from 7^23 up to 7^33 are:
IntSqrt(7^23)=5231514822 of 7^23 (27368747340080916343)
IntSqrt(7^25)=36620603758 of 7^25 (1341068619663964900807)
IntSqrt(7^27)=256344226312 of 7^27 (65712362363534280139543)
IntSqrt(7^29)=1794409584184 of 7^29 (3219905755813179726837607)
IntSqrt(7^31)=12560867089291 of 7^31 (157775382034845806615042743)
IntSqrt(7^33)=87926069625040 of 7^33 (7730993719707444524137094407)

Using Double Type
The integer square roots of powers of 7 from 7^19 up to 7^35 are:
IntSqrt(7^19)=106765608 of 7^19 (1.13988951853731E+16)
IntSqrt(7^21)=747359260 of 7^21 (5.58545864083284E+17)
IntSqrt(7^23)=5231514822 of 7^23 (2.73687473400809E+19)
IntSqrt(7^25)=36620603758 of 7^25 (1.34106861966396E+21)
IntSqrt(7^27)=256344226312 of 7^27 (6.57123623635343E+22)
IntSqrt(7^29)=1794409584184 of 7^29 (3.21990575581318E+24)
IntSqrt(7^31)=12560867089291 of 7^31 (1.57775382034846E+26)
IntSqrt(7^33)=87926069625040 of 7^33 (7.73099371970744E+27)
IntSqrt(7^35)=615482487375282 of 7^35 (3.78818692265665E+29)

```

```            NORMAL MODE IS INTEGER

R  INTEGER SQUARE ROOT OF X

INTERNAL FUNCTION(X)
ENTRY TO ISQRT.
Q = 1
FNDPW4      WHENEVER Q.LE.X
Q = Q * 4
TRANSFER TO FNDPW4
END OF CONDITIONAL
Z = X
R = 0
FNDRT       WHENEVER Q.G.1
Q = Q / 4
T = Z - R - Q
R = R / 2
WHENEVER T.GE.0
Z = T
R = R + Q
END OF CONDITIONAL
TRANSFER TO FNDRT
END OF CONDITIONAL
FUNCTION RETURN R
END OF FUNCTION

R  PRINT INTEGER SQUARE ROOTS OF 0..65

THROUGH SQ65, FOR N=0, 11, N.G.65
SQ65        PRINT FORMAT N11, ISQRT.(N), ISQRT.(N+1), ISQRT.(N+2),
0    ISQRT.(N+3), ISQRT.(N+4), ISQRT.(N+5), ISQRT.(N+6),
1    ISQRT.(N+7), ISQRT.(N+8), ISQRT.(N+9), ISQRT.(N+10)
VECTOR VALUES N11 = \$11(I1,S1)*\$

R  MACHINE WORD SIZE ON IBM 704 IS 36 BITS
R  PRINT UP TO AND INCLUDING ISQRT(7^12)

POW7 = 1
THROUGH SQ7P12, FOR I=0, 1, I.G.12
PRINT FORMAT SQ7, I, ISQRT.(POW7)
SQ7P12      POW7 = POW7 * 7
VECTOR VALUES SQ7 = \$9HISQRT.(7^,I2,4H) = ,I6*\$

END OF PROGRAM```
Output:
```0 1 1 1 2 2 2 2 2 3 3
3 3 3 3 3 4 4 4 4 4 4
4 4 4 5 5 5 5 5 5 5 5
5 5 5 6 6 6 6 6 6 6 6
6 6 6 6 6 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 8 8
ISQRT.(7^ 0) =      1
ISQRT.(7^ 1) =      2
ISQRT.(7^ 2) =      7
ISQRT.(7^ 3) =     18
ISQRT.(7^ 4) =     49
ISQRT.(7^ 5) =    129
ISQRT.(7^ 6) =    343
ISQRT.(7^ 7) =    907
ISQRT.(7^ 8) =   2401
ISQRT.(7^ 9) =   6352
ISQRT.(7^10) =  16807
ISQRT.(7^11) =  44467
ISQRT.(7^12) = 117649```

## Mathematica/Wolfram Language

```ClearAll[ISqrt]
ISqrt[x_Integer?NonNegative] := Module[{q = 1, z, r, t},
While[q <= x,
q *= 4
];
z = x;
r = 0;
While[q > 1,
q = Quotient[q, 4];
t = z - r - q;
r /= 2;
If[t >= 0,
z = t;
r += q
];
];
r
]
ISqrt /@ Range[65]
Column[ISqrt /@ (7^Range[1, 73])]
```
Output:
```{1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8}
2
7
18
49
129
343
907
2401
6352
16807
44467
117649
311269
823543
2178889
5764801
15252229
40353607
106765608
282475249
747359260
1977326743
5231514822
13841287201
36620603758
96889010407
256344226312
678223072849
1794409584184
4747561509943
12560867089291
33232930569601
87926069625040
232630513987207
615482487375282
1628413597910449
4308377411626977
11398895185373143
30158641881388842
79792266297612001
211110493169721897
558545864083284007
1477773452188053281
3909821048582988049
10344414165316372973
27368747340080916343
72410899157214610812
191581231380566414401
506876294100502275687
1341068619663964900807
3548134058703515929815
9387480337647754305649
24836938410924611508707
65712362363534280139543
173858568876472280560953
459986536544739960976801
1217009982135305963926677
3219905755813179726837607
8519069874947141747486745
22539340290692258087863249
59633489124629992232407216
157775382034845806615042743
417434423872409945626850517
1104427674243920646305299201
2922040967106869619387953625
7730993719707444524137094407
20454286769748087335715675381
54116956037952111668959660849
143180007388236611350009727669
378818692265664781682717625943
1002260051717656279450068093686
2651730845859653471779023381601
7015820362023593956150476655802```

## Maxima

```/* -*- Maxima -*- */

/*

The Rosetta Code integer square root task, in Maxima.

I have not tried to make the output conform quite to the task
description, because Maxima is not a general purpose programming
language. Perhaps someone else will care to do it.

I *do* check that the Rosetta Code routine gives the same results as
the built-in function.

*/

/* pow4gtx -- find a power of 4 greater than x. */
pow4gtx (x) := block (
[q],
q : 1, while q <= x do q : bit_lsh (q, 2),
q
) \$

/* rosetta_code_isqrt -- find the integer square root. */
rosetta_code_isqrt (x) := block (
[q, z, r, t],
q : pow4gtx (x),
z : x,
r : 0,
while 1 < q do (
q : bit_rsh (q, 2),
t : z - r - q,
r : bit_rsh (r, 1),
if 0 <= t then (
z : t,
r : r + q
)
),
r
) \$

for i : 0 thru 65 do (
display (rosetta_code_isqrt (i),
is (equal (rosetta_code_isqrt (i), isqrt (i))))
) \$
for i : 1 thru 73 step 2 do (
display (7**i, rosetta_code_isqrt (7**i),
is (equal (rosetta_code_isqrt (7**i), isqrt (7**i))))
) \$
```
Output:
```\$ maxima -q -b isqrt.mac
(%i1) batch("isqrt.mac")

(%i2) pow4gtx(x):=block([q],q:1,while q <= x do q:bit_lsh(q,2),q)
(%i3) rosetta_code_isqrt(x):=block([q,z,r,t],q:pow4gtx(x),z:x,r:0,
while 1 < q do
(q:bit_rsh(q,2),t:z-r-q,r:bit_rsh(r,1),
if 0 <= t then (z:t,r:r+q)),r)
(%i4) for i from 0 thru 65 do
display(rosetta_code_isqrt(i),
is(equal(rosetta_code_isqrt(i),isqrt(i))))
rosetta_code_isqrt(0) = 0

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(1) = 1

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(2) = 1

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(3) = 1

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(4) = 2

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(5) = 2

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(6) = 2

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(7) = 2

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(8) = 2

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(9) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(10) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(11) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(12) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(13) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(14) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(15) = 3

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(16) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(17) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(18) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(19) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(20) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(21) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(22) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(23) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(24) = 4

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(25) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(26) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(27) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(28) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(29) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(30) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(31) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(32) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(33) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(34) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(35) = 5

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(36) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(37) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(38) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(39) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(40) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(41) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(42) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(43) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(44) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(45) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(46) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(47) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(48) = 6

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(49) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(50) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(51) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(52) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(53) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(54) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(55) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(56) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(57) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(58) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(59) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(60) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(61) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(62) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(63) = 7

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(64) = 8

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

rosetta_code_isqrt(65) = 8

is(equal(rosetta_code_isqrt(i), isqrt(i))) = true

(%i5) for i step 2 thru 73 do
display(7^i,rosetta_code_isqrt(7^i),
is(equal(rosetta_code_isqrt(7^i),isqrt(7^i))))
1
7  = 7

rosetta_code_isqrt(7) = 2

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

3
7  = 343

rosetta_code_isqrt(343) = 18

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

5
7  = 16807

rosetta_code_isqrt(16807) = 129

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

7
7  = 823543

rosetta_code_isqrt(823543) = 907

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

9
7  = 40353607

rosetta_code_isqrt(40353607) = 6352

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

11
7   = 1977326743

rosetta_code_isqrt(1977326743) = 44467

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

13
7   = 96889010407

rosetta_code_isqrt(96889010407) = 311269

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

15
7   = 4747561509943

rosetta_code_isqrt(4747561509943) = 2178889

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

17
7   = 232630513987207

rosetta_code_isqrt(232630513987207) = 15252229

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

19
7   = 11398895185373143

rosetta_code_isqrt(11398895185373143) = 106765608

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

21
7   = 558545864083284007

rosetta_code_isqrt(558545864083284007) = 747359260

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

23
7   = 27368747340080916343

rosetta_code_isqrt(27368747340080916343) = 5231514822

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

25
7   = 1341068619663964900807

rosetta_code_isqrt(1341068619663964900807) = 36620603758

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

27
7   = 65712362363534280139543

rosetta_code_isqrt(65712362363534280139543) = 256344226312

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

29
7   = 3219905755813179726837607

rosetta_code_isqrt(3219905755813179726837607) = 1794409584184

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

31
7   = 157775382034845806615042743

rosetta_code_isqrt(157775382034845806615042743) = 12560867089291

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

33
7   = 7730993719707444524137094407

rosetta_code_isqrt(7730993719707444524137094407) = 87926069625040

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

35
7   = 378818692265664781682717625943

rosetta_code_isqrt(378818692265664781682717625943) = 615482487375282

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

37
7   = 18562115921017574302453163671207

rosetta_code_isqrt(18562115921017574302453163671207) = 4308377411626977

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

39
7   = 909543680129861140820205019889143

rosetta_code_isqrt(909543680129861140820205019889143) = 30158641881388842

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

41
7   = 44567640326363195900190045974568007

rosetta_code_isqrt(44567640326363195900190045974568007) = 211110493169721897

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

43
7   = 2183814375991796599109312252753832343

rosetta_code_isqrt(2183814375991796599109312252753832343) = 1477773452188053281

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

45
7   = 107006904423598033356356300384937784807

rosetta_code_isqrt(107006904423598033356356300384937784807) =
10344414165316372973

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

47
7   = 5243338316756303634461458718861951455543

rosetta_code_isqrt(5243338316756303634461458718861951455543) =
72410899157214610812

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

49
7   = 256923577521058878088611477224235621321607

rosetta_code_isqrt(256923577521058878088611477224235621321607) =
506876294100502275687

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

51
7   = 12589255298531885026341962383987545444758743

rosetta_code_isqrt(12589255298531885026341962383987545444758743) =
3548134058703515929815

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

53
7   = 616873509628062366290756156815389726793178407

rosetta_code_isqrt(616873509628062366290756156815389726793178407) =
24836938410924611508707

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

55
7   = 30226801971775055948247051683954096612865741943

rosetta_code_isqrt(30226801971775055948247051683954096612865741943) =
173858568876472280560953

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

57
7   = 1481113296616977741464105532513750734030421355207

rosetta_code_isqrt(1481113296616977741464105532513750734030421355207) =
1217009982135305963926677

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

59
7   = 72574551534231909331741171093173785967490646405143

rosetta_code_isqrt(72574551534231909331741171093173785967490646405143) =
8519069874947141747486745

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

61
7   = 3556153025177363557255317383565515512407041673852007

rosetta_code_isqrt(3556153025177363557255317383565515512407041673852007) =
59633489124629992232407216

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

63
7   = 174251498233690814305510551794710260107945042018748343

rosetta_code_isqrt(174251498233690814305510551794710260107945042018748343) =
417434423872409945626850517

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

65
7   = 8538323413450849900970017037940802745289307058918668807

rosetta_code_isqrt(8538323413450849900970017037940802745289307058918668807) =
2922040967106869619387953625

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

67
7   = 418377847259091645147530834859099334519176045887014771543

rosetta_code_isqrt(418377847259091645147530834859099334519176045887014771543
) = 20454286769748087335715675381

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

69
7   = 20500514515695490612229010908095867391439626248463723805607

rosetta_code_isqrt(20500514515695490612229010908095867391439626248463723805607
) = 143180007388236611350009727669

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

71
7   = 1004525211269079039999221534496697502180541686174722466474743

rosetta_code_isqrt(10045252112690790399992215344966975021805416861747224664747\
43) = 1002260051717656279450068093686

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

73
7   = 49221735352184872959961855190338177606846542622561400857262407

rosetta_code_isqrt(49221735352184872959961855190338177606846542622561400857262\
407) = 7015820362023593956150476655802

i          i
is(equal(rosetta_code_isqrt(7 ), isqrt(7 ))) = true

(%o6) /home/trashman/src/chemoelectric/rosettacode-contributions/isqrt.mac```

## Mercury

Translation of: Prolog
Works with: Mercury version 20.06.1

```:- module isqrt_in_mercury.

:- interface.
:- import_module io.
:- pred main(io, io).
:- mode main(di, uo) is det.

:- implementation.
:- import_module char.
:- import_module exception.
:- import_module int.
:- import_module integer.       % Integers of arbitrary size.
:- import_module list.
:- import_module string.

:- func four = integer.
four = integer(4).

:- func seven = integer.
seven = integer(7).

%% Find a power of 4 greater than X.
:- func pow4gtx(integer) = integer.
pow4gtx(X) = Q :- pow4gtx_(X, one, Q).

%% The tail recursion for pow4gtx.
:- pred pow4gtx_(integer, integer, integer).
:- mode pow4gtx_(in, in, out) is det.
pow4gtx_(X, A, Q) :- if (X < A) then (Q = A)
else (A1 = A * four,
pow4gtx_(X, A1, Q)).

%% Integer square root function.
:- func isqrt(integer) = integer.
isqrt(X) = Root :- isqrt(X, Root, _).

%% Integer square root and remainder.
:- pred isqrt(integer, integer, integer).
:- mode isqrt(in, out, out) is det.
isqrt(X, Root, Remainder) :-
Q = pow4gtx(X),
isqrt_(X, Q, zero, X, Root, Remainder).

%% The tail recursion for isqrt.
:- pred isqrt_(integer, integer, integer, integer, integer, integer).
:- mode isqrt_(in, in, in, in, out, out) is det.
isqrt_(X, Q, R0, Z0, R, Z) :-
if (X < zero) then throw("isqrt of a negative integer")
else if (Q = one) then (R = R0, Z = Z0)
else (Q1 = Q // four,
T = Z0 - R0 - Q1,
(if (T >= zero)
then (R1 = (R0 // two) + Q1,
isqrt_(X, Q1, R1, T, R, Z))
else (R1 is R0 // two,
isqrt_(X, Q1, R1, Z0, R, Z)))).

%% Insert a character, every three digits, into (what presumably is)
%% an integer numeral. (The name "commas" is not very good.)
:- func commas(string, char) = string.
commas(S, Comma) = T :-
RCL = to_rev_char_list(S),
commas_(RCL, Comma, 0, [], CL),
T = from_char_list(CL).

%% The tail recursion for commas.
:- pred commas_(list(char), char, int, list(char), list(char)).
:- mode commas_(in, in, in, in, out) is det.
commas_([], _, _, L, CL) :- L = CL.
commas_([C | Tail], Comma, I, L, CL) :-
if (I = 3) then commas_([C | Tail], Comma, 0, [Comma | L], CL)
else (I1 = I + 1,
commas_(Tail, Comma, I1, [C | L], CL)).

:- pred roots_m_to_n(integer, integer, io, io).
:- mode roots_m_to_n(in, in, di, uo) is det.
roots_m_to_n(M, N, !IO) :-
if (N < M) then true
else (write_string(to_string(isqrt(M)), !IO),
(if (M \= N) then write_string(" ", !IO)
else true),
M1 = M + one,
roots_m_to_n(M1, N, !IO)).

:- pred roots_of_odd_powers_of_7(integer, integer, io, io).
:- mode roots_of_odd_powers_of_7(in, in, di, uo) is det.
roots_of_odd_powers_of_7(M, N, !IO) :-
if (N < M) then true
else (Pow7 = pow(seven, M),
Isqrt = isqrt(Pow7),
format("%2s %84s %43s",
[s(commas(to_string(M), (','))),
s(commas(to_string(Pow7), (','))),
s(commas(to_string(Isqrt), (',')))],
!IO),
nl(!IO),
M1 = M + two,
roots_of_odd_powers_of_7(M1, N, !IO)).

main(!IO) :-
write_string("isqrt(i) for 0 <= i <= 65:", !IO),
nl(!IO), nl(!IO),
roots_m_to_n(zero, integer(65), !IO),
nl(!IO), nl(!IO), nl(!IO),
write_string("isqrt(7**i) for 1 <= i <= 73, i odd:", !IO),
nl(!IO), nl(!IO),
format("%2s %84s %43s", [s("i"), s("7**i"), s("isqrt(7**i)")], !IO),
nl(!IO),
write_string(duplicate_char(('-'), 131), !IO),
nl(!IO),
roots_of_odd_powers_of_7(one, integer(73), !IO).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Instructions for GNU Emacs--
%%% local variables:
%%% mode: mercury
%%% prolog-indent-width: 2
%%% end:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%```
Output:
```\$ mmc --warn-non-tail-recursion=self-and-mutual -o isqrt isqrt_in_mercury.m && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                 isqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## Modula-2

Uses the algorithm in the task description, with two modifications: (1) As noted in the ALGOL-M solution, the original algorithm can lead to integer overflow when trying to find a power of 4 greater than the argument X. The modified algorithm avoids overflow. (2) In the original algorithm, the variable t can be negative. In the modified algorithm, all variables remain non-negative, and can therefore be declared as unsigned integers if desired.

TopSpeed Modula-2 supports no integers larger than unsigned 32-bit, which means that the second part of the task stops at 7^11. There seems to be no option to insert commas into long numbers as requested.

```MODULE IntSqrt;

IMPORT IO;

(* Procedure to find integer square root of a 32-bit unsigned integer. *)
PROCEDURE Isqrt( X : LONGCARD) : LONGCARD;
VAR
Xdiv4, q, r, s, z : LONGCARD;
BEGIN
Xdiv4 := X DIV 4;
q := 1;
WHILE q <= Xdiv4 DO q := 4*q; END;
z := X;
r := 0;
REPEAT
s := q + r;
r := r DIV 2;
IF z >= s THEN
DEC(z, s);
INC(r, q);
END;
q := q DIV 4;
UNTIL q = 0;
RETURN r;
END Isqrt;

(* Main program *)
CONST (* constants for Part 1 of the task *)
Max_n = 65;
NrPerLine = 22;
VAR
n : LONGCARD;
arr_n, arr_i : ARRAY[0..NrPerLine - 1] OF LONGCARD; (* for display *)
j, k : INTEGER;
BEGIN
(* Part 1 *)
k := 0; (* index into arrays *)
FOR n := 0 TO Max_n DO
arr_n[k] := n;
arr_i[k] := Isqrt(n);
INC(k);
IF (k = NrPerLine) OR (n = Max_n) THEN
IO.WrStr( 'Number: ');
FOR j := 0 TO k - 1 DO IO.WrLngCard(arr_n[j], 3); END;
IO.WrLn();
IO.WrStr( 'Isqrt:  ');
FOR j := 0 TO k - 1 DO IO.WrLngCard(arr_i[j], 3); END;
IO.WrLn();
k := 0;
END;
END;
IO.WrLn();

(* Part 2 *)
IO.WrStr( 'Isqrt of odd powers of 7'); IO.WrLn();
n := 7;
FOR k := 1 TO 11 BY 2 DO
IF k > 1 THEN n := n*49; END;
IO.WrInt( k, 2); IO.WrStr( ' --> ');
IO.WrLngCard( n, 10); IO.WrStr(' --> ');
IO.WrLngCard( Isqrt(n), 5); IO.WrLn();
END;
END IntSqrt.
```
Output:
```Number:   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21
Isqrt:    0  1  1  1  2  2  2  2  2  3  3  3  3  3  3  3  4  4  4  4  4  4
Number:  22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
Isqrt:    4  4  4  5  5  5  5  5  5  5  5  5  5  5  6  6  6  6  6  6  6  6
Number:  44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Isqrt:    6  6  6  6  6  7  7  7  7  7  7  7  7  7  7  7  7  7  7  7  8  8

Isqrt of odd powers of 7
1 -->          7 -->     2
3 -->        343 -->    18
5 -->      16807 -->   129
7 -->     823543 -->   907
9 -->   40353607 -->  6352
11 --> 1977326743 --> 44467
```

## Nim

Library: bignum

This Nim implementation provides an `isqrt` function for signed integers and for big integers.

```import strformat, strutils
import bignum

func isqrt*[T: SomeSignedInt | Int](x: T): T =
## Compute integer square root for signed integers
## and for big integers.

when T is Int:
result = newInt()
var q = newInt(1)
else:
result = 0
var q = T(1)

while q <= x:
q = q shl 2

var z = x
while q > 1:
q = q shr 2
let t = z - result - q
result = result shr 1
if t >= 0:
z = t
result += q

when isMainModule:

echo "Integer square root for numbers 0 to 65:"
for n in 0..65:
stdout.write ' ', isqrt(n)
echo "\n"

echo "Integer square roots of odd powers of 7 from 7^1 to 7^73:"
echo " n" & repeat(' ', 82) & "7^n" & repeat(' ', 34) & "isqrt(7^n)"
echo repeat("—", 131)

var x = newInt(7)
for n in countup(1, 73, 2):
echo &"{n:>2}   {insertSep(\$x, ','):>82}   {insertSep(\$isqrt(x), ','):>41}"
x *= 49
```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 7^1 to 7^73:
n                                                                                  7^n                                  isqrt(7^n)
———————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## ObjectIcon

Translation of: Icon

The only changes needed to make the Icon version work in Object Icon were to "import io" and to import the IPL modules in a different way. ("write" and "writes" are supported by the io module, presumably for compatibility with Icon and to ease migration of the IPL.)

```# -*- ObjectIcon -*-

import io
import ipl.numbers              # For the "commas" procedure.
import ipl.printf

procedure main ()
write ("isqrt(i) for 0 <= i <= 65:")
write ()
roots_of_0_to_65()
write ()
write ()
write ("isqrt(7**i) for 1 <= i <= 73, i odd:")
write ()
printf ("%2s %84s %43s\n", "i", "7**i", "sqrt(7**i)")
write (repl("-", 131))
roots_of_odd_powers_of_7()
end

procedure roots_of_0_to_65 ()
local i

every i := 0 to 64 do writes (isqrt(i), " ")
write (isqrt(65))
end

procedure roots_of_odd_powers_of_7 ()
local i, power_of_7, root

every i := 1 to 73 by 2 do {
power_of_7 := 7^i
root := isqrt(power_of_7)
printf ("%2d %84s %43s\n", i, commas(power_of_7), commas(root))
}
end

procedure isqrt (x)
local q, z, r, t

q := find_a_power_of_4_greater_than_x (x)
z := x
r := 0
while 1 < q do {
q := ishift(q, -2)
t := z - r - q
r := ishift(r, -1)
if 0 <= t then {
z := t
r +:= q
}
}
return r
end

procedure find_a_power_of_4_greater_than_x (x)
local q

q := 1
while q <= x do q := ishift(q, 2)
return q
end```
Output:
```\$ oit -s -o isqrt isqrt-in-OI.icn && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## OCaml

Translation of: Scheme
Library: Zarith

```(* The Rosetta Code integer square root task, in OCaml, using Zarith
for large integers.

Compile with, for example:

ocamlfind ocamlc -package zarith -linkpkg -o isqrt isqrt.ml

Translated from the Scheme. *)

let find_a_power_of_4_greater_than_x x =
let open Z in
let rec loop q =
if x < q then q else loop (q lsl 2)
in
loop one

let isqrt x =
let open Z in
let rec loop q z r =
if q = one then
r
else
let q = q asr 2 in
let t = z - r - q in
let r = r asr 1 in
if t < zero then
loop q z r
else
loop q t (r + q)
in
let q0 = find_a_power_of_4_greater_than_x x in
let z0 = x in
let r0 = zero in
loop q0 z0 r0

let insert_separators s sep =
let rec loop revchars i newchars =
match revchars with
| [] -> newchars
| revchars when i = 3 -> loop revchars 0 (sep :: newchars)
| c :: tail -> loop tail (i + 1) (c :: newchars)
in
let revchars = List.rev (List.of_seq (String.to_seq s)) in
String.of_seq (List.to_seq (loop revchars 0 []))

let commas s = insert_separators s ','

let main () =
Printf.printf "isqrt(i) for 0 <= i <= 65:\n\n";
for i = 0 to 64 do
Printf.printf "%s " Z.(to_string (isqrt (of_int i)))
done;
Printf.printf "%s\n" Z.(to_string (isqrt (of_int 65)));
Printf.printf "\n\n";
Printf.printf "isqrt(7**i) for 1 <= i <= 73, i odd:\n\n";
Printf.printf "%2s %84s %43s\n" "i" "7**i" "isqrt(7**i)";
for i = 1 to 131 do Printf.printf "-" done;
Printf.printf "\n";
for j = 0 to 36 do
let i = j + j + 1 in
let power = Z.(of_int 7 ** i) in
let root = isqrt power in
Printf.printf "%2d %84s %43s\n"
i (commas (Z.to_string power)) (commas (Z.to_string root))
done
;;

main ()
```
Output:
```\$ ocamlfind ocamlc -package zarith -linkpkg -o isqrt isqrt.ml && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                 isqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## Ol

```(print "Integer square roots of 0..65")
(for-each (lambda (x)
(display (isqrt x))
(display " "))
(iota 66))
(print)

(print "Integer square roots of 7^n")
(for-each (lambda (x)
(print "x: " x ", isqrt: " (isqrt x)))
(map (lambda (i)
(expt 7 i))
(iota 73 1)))
(print)
```
Output:
```Integer square roots of 0..65
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Integer square roots of 7^n
x: 7, isqrt: 2
x: 49, isqrt: 7
x: 343, isqrt: 18
x: 2401, isqrt: 49
x: 16807, isqrt: 129
x: 117649, isqrt: 343
x: 823543, isqrt: 907
x: 5764801, isqrt: 2401
x: 40353607, isqrt: 6352
x: 282475249, isqrt: 16807
x: 1977326743, isqrt: 44467
x: 13841287201, isqrt: 117649
x: 96889010407, isqrt: 311269
x: 678223072849, isqrt: 823543
x: 4747561509943, isqrt: 2178889
x: 33232930569601, isqrt: 5764801
x: 232630513987207, isqrt: 15252229
x: 1628413597910449, isqrt: 40353607
x: 11398895185373143, isqrt: 106765608
x: 79792266297612001, isqrt: 282475249
x: 558545864083284007, isqrt: 747359260
x: 3909821048582988049, isqrt: 1977326743
x: 27368747340080916343, isqrt: 5231514822
x: 191581231380566414401, isqrt: 13841287201
x: 1341068619663964900807, isqrt: 36620603758
x: 9387480337647754305649, isqrt: 96889010407
x: 65712362363534280139543, isqrt: 256344226312
x: 459986536544739960976801, isqrt: 678223072849
x: 3219905755813179726837607, isqrt: 1794409584184
x: 22539340290692258087863249, isqrt: 4747561509943
x: 157775382034845806615042743, isqrt: 12560867089291
x: 1104427674243920646305299201, isqrt: 33232930569601
x: 7730993719707444524137094407, isqrt: 87926069625040
x: 54116956037952111668959660849, isqrt: 232630513987207
x: 378818692265664781682717625943, isqrt: 615482487375282
x: 2651730845859653471779023381601, isqrt: 1628413597910449
x: 18562115921017574302453163671207, isqrt: 4308377411626977
x: 129934811447123020117172145698449, isqrt: 11398895185373143
x: 909543680129861140820205019889143, isqrt: 30158641881388842
x: 6366805760909027985741435139224001, isqrt: 79792266297612001
x: 44567640326363195900190045974568007, isqrt: 211110493169721897
x: 311973482284542371301330321821976049, isqrt: 558545864083284007
x: 2183814375991796599109312252753832343, isqrt: 1477773452188053281
x: 15286700631942576193765185769276826401, isqrt: 3909821048582988049
x: 107006904423598033356356300384937784807, isqrt: 10344414165316372973
x: 749048330965186233494494102694564493649, isqrt: 27368747340080916343
x: 5243338316756303634461458718861951455543, isqrt: 72410899157214610812
x: 36703368217294125441230211032033660188801, isqrt: 191581231380566414401
x: 256923577521058878088611477224235621321607, isqrt: 506876294100502275687
x: 1798465042647412146620280340569649349251249, isqrt: 1341068619663964900807
x: 12589255298531885026341962383987545444758743, isqrt: 3548134058703515929815
x: 88124787089723195184393736687912818113311201, isqrt: 9387480337647754305649
x: 616873509628062366290756156815389726793178407, isqrt: 24836938410924611508707
x: 4318114567396436564035293097707728087552248849, isqrt: 65712362363534280139543
x: 30226801971775055948247051683954096612865741943, isqrt: 173858568876472280560953
x: 211587613802425391637729361787678676290060193601, isqrt: 459986536544739960976801
x: 1481113296616977741464105532513750734030421355207, isqrt: 1217009982135305963926677
x: 10367793076318844190248738727596255138212949486449, isqrt: 3219905755813179726837607
x: 72574551534231909331741171093173785967490646405143, isqrt: 8519069874947141747486745
x: 508021860739623365322188197652216501772434524836001, isqrt: 22539340290692258087863249
x: 3556153025177363557255317383565515512407041673852007, isqrt: 59633489124629992232407216
x: 24893071176241544900787221684958608586849291716964049, isqrt: 157775382034845806615042743
x: 174251498233690814305510551794710260107945042018748343, isqrt: 417434423872409945626850517
x: 1219760487635835700138573862562971820755615294131238401, isqrt: 1104427674243920646305299201
x: 8538323413450849900970017037940802745289307058918668807, isqrt: 2922040967106869619387953625
x: 59768263894155949306790119265585619217025149412430681649, isqrt: 7730993719707444524137094407
x: 418377847259091645147530834859099334519176045887014771543, isqrt: 20454286769748087335715675381
x: 2928644930813641516032715844013695341634232321209103400801, isqrt: 54116956037952111668959660849
x: 20500514515695490612229010908095867391439626248463723805607, isqrt: 143180007388236611350009727669
x: 143503601609868434285603076356671071740077383739246066639249, isqrt: 378818692265664781682717625943
x: 1004525211269079039999221534496697502180541686174722466474743, isqrt: 1002260051717656279450068093686
x: 7031676478883553279994550741476882515263791803223057265323201, isqrt: 2651730845859653471779023381601
x: 49221735352184872959961855190338177606846542622561400857262407, isqrt: 7015820362023593956150476655802
```

## Pascal

[1]
Translation of: C++
```//************************************************//
//                                                //
//  Thanks for rvelthuis for BigIntegers library  //
//  https://github.com/rvelthuis/DelphiBigNumbers //
//                                                //
//************************************************//

{\$APPTYPE CONSOLE}

{\$R *.res}

uses
System.SysUtils,
Velthuis.BigIntegers;

function isqrt(x: BigInteger): BigInteger;
var
q, r, t: BigInteger;
begin
q := 1;
r := 0;
while (q <= x) do
q := q shl 2;

while (q > 1) do
begin
q := q shr 2;
t := x - r - q;
r := r shr 1;
if (t >= 0) then
begin
x := t;
r := r + q;
end;
end;
Result := r;
end;

function commatize(const n: BigInteger; size: Integer): string;
var
str: string;
digits: Integer;
i: Integer;
begin
Result := '';
str := n.ToString;
digits := str.Length;

for i := 1 to digits do
begin
if ((i > 1) and (((i - 1) mod 3) = (digits mod 3))) then
Result := Result + ',';
Result := Result + str[i];
end;

if Result.Length < size then
Result := string.Create(' ', size - Result.Length) + Result;
end;

const
POWER_WIDTH = 83;
ISQRT_WIDTH = 42;

var
n, i: Integer;
f: TextFile;
p: BigInteger;

begin
AssignFile(f, 'output.txt');
rewrite(f);

Writeln(f, 'Integer square root for numbers 0 to 65:');
for n := 0 to 65 do
Write(f, isqrt(n).ToString, ' ');

Writeln(f, #10#10'Integer square roots of odd powers of 7 from 1 to 73:');

Write(f, ' n |', string.Create(' ', 78), '7 ^ n |', string.Create(' ', 30),
'isqrt(7 ^ n)'#10);

Writeln(f, string.Create('-', 17 + POWER_WIDTH + ISQRT_WIDTH));

p := 7;
n := 1;
repeat
Writeln(f, Format('%2d', [n]), ' |', commatize(p, power_width), ' |',
commatize(isqrt(p), isqrt_width));
inc(n, 2);
p := p * 49;
until (n > 73);

CloseFile(f);
end.
```

## Perl

Translation of: Julia
```# 20201029 added Perl programming solution

use strict;
use warnings;
use bigint;

use CLDR::Number 'decimal_formatter';

sub integer_sqrt {
( my \$x = \$_[0] ) >= 0 or die;
my \$q = 1;
while (\$q <= \$x) {
\$q <<= 2
}
my (\$z, \$r) = (\$x, 0);
while (\$q > 1) {
\$q >>= 2;
my \$t = \$z - \$r - \$q;
\$r >>= 1;
if (\$t >= 0) {
\$z  = \$t;
\$r += \$q;
}
}
return \$r
}

print "The integer square roots of integers from 0 to 65 are:\n";
print map { ( integer_sqrt \$_ ) . ' ' } (0..65);

my \$cldr = CLDR::Number->new();
my \$decf = \$cldr->decimal_formatter;

print "\nThe integer square roots of odd powers of 7 from 7^1 up to 7^73 are:\n";
print "power", " "x36, "7 ^ power", " "x60, "integer square root\n";
print "----- ", "-"x79, "  ------------------------------------------\n";

for (my \$i = 1; \$i < 74; \$i += 2) {
printf("%2s ", \$i);
printf("%82s", \$decf->format( 7**\$i ) );
printf("%44s", \$decf->format( integer_sqrt(7**\$i) ) ) ;
print "\n";
}
```
Output:
```The integer square roots of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:
power                                    7 ^ power                                                            integer square root
----- -------------------------------------------------------------------------------  ------------------------------------------
1                                                                                  7                                           2
3                                                                                343                                          18
5                                                                             16,807                                         129
7                                                                            823,543                                         907
9                                                                         40,353,607                                       6,352
11                                                                      1,977,326,743                                      44,467
13                                                                     96,889,010,407                                     311,269
15                                                                  4,747,561,509,943                                   2,178,889
17                                                                232,630,513,987,207                                  15,252,229
19                                                             11,398,895,185,373,143                                 106,765,608
21                                                            558,545,864,083,284,007                                 747,359,260
23                                                         27,368,747,340,080,916,343                               5,231,514,822
25                                                      1,341,068,619,663,964,900,807                              36,620,603,758
27                                                     65,712,362,363,534,280,139,543                             256,344,226,312
29                                                  3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                              7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                            378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                         18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                        909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                     44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                  2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                              5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                            256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802
```

## Phix

See also Integer_roots#Phix for a simpler and shorter example using the mpz_root() routine, or better yet just use mpz_root() directly (that is, rather than the isqrt() below).

```with javascript_semantics
include mpfr.e

function isqrt(mpz x)
if mpz_cmp_si(x,0)<0 then
crash("Argument cannot be negative.")
end if
mpz q = mpz_init(1),
r = mpz_init(0),
t = mpz_init(),
z = mpz_init_set(x)
while mpz_cmp(q,x)<= 0 do
mpz_mul_si(q,q,4)
end while
while mpz_cmp_si(q,1)>0 do
assert(mpz_fdiv_q_ui(q, q, 4)=0)
mpz_sub(t,z,r)
mpz_sub(t,t,q)
assert(mpz_fdiv_q_ui(r, r, 2)=0)
if mpz_cmp_si(t,0) >= 0 then
mpz_set(z,t)
end if
end while
string star = iff(mpz_cmp_si(z,0)=0?"*":" ")
return shorten(mpz_get_str(r,10,true))&star
end function

printf(1,"The integer square roots of integers from 0 to 65 are:\n")
for i=0 to 65 do
printf(1,"%s ", {trim(isqrt(mpz_init(i)))})
end for
printf(1,"\n\npower                          7 ^ power                                               integer square root\n")
printf(1,"-----  ---------------------------------------------------------   ----------------------------------------------------------\n")
mpz pow7 = mpz_init(7)
for i=1 to 9000 do
if (i<=73  and remainder(i,2)=1)
or (i<100  and remainder(i,10)=5)
or (i<1000 and remainder(i,100)=0)
or             remainder(i,1000)=0 then
printf(1,"%4d  %58s %61s\n", {i, shorten(mpz_get_str(pow7,10,true)),isqrt(pow7)})
end if
mpz_mul_si(pow7,pow7,7)
end for
```
Output:

Perfect squares are denoted with an asterisk.

```The integer square roots of integers from 0 to 65 are:
0* 1* 1 1 2* 2 2 2 2 3* 3 3 3 3 3 3 4* 4 4 4 4 4 4 4 4 5* 5 5 5 5 5 5 5 5 5 5 6* 6 6 6 6 6 6 6 6 6 6 6 6 7* 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8* 8

power                          7 ^ power                                               integer square root
-----  ---------------------------------------------------------   ----------------------------------------------------------
1                                                           7                                                            2
3                                                         343                                                           18
5                                                      16,807                                                          129
7                                                     823,543                                                          907
9                                                  40,353,607                                                        6,352
11                                               1,977,326,743                                                       44,467
13                                              96,889,010,407                                                      311,269
15                                           4,747,561,509,943                                                    2,178,889
17                                         232,630,513,987,207                                                   15,252,229
19                                      11,398,895,185,373,143                                                  106,765,608
21                                     558,545,864,083,284,007                                                  747,359,260
23                                  27,368,747,340,080,916,343                                                5,231,514,822
25                               1,341,068,619,663,964,900,807                                               36,620,603,758
27                              65,712,362,363,534,280,139,543                                              256,344,226,312
29                           3,219,905,755,813,179,726,837,607                                            1,794,409,584,184
31                         157,775,382,034,845,806,615,042,743                                           12,560,867,089,291
33                       7,730,993,719,707,444,524,137,094,407                                           87,926,069,625,040
35                     378,818,692,265,664,781,682,717,625,943                                          615,482,487,375,282
37                  18,562,115,921,017,574,302,453,163,671,207                                        4,308,377,411,626,977
39                 909,543,680,129,861,140,820,205,019,889,143                                       30,158,641,881,388,842
41              44,567,640,326,363,195,900,190,045,974,568,007                                      211,110,493,169,721,897
43           2,183,814,375,991,796,599,109,312,252,753,832,343                                    1,477,773,452,188,053,281
45         107,006,904,423,598,033,356,356,300,384,937,784,807                                   10,344,414,165,316,372,973
47       5,243,338,316,756,303,634,461,458,718,861,951,455,543                                   72,410,899,157,214,610,812
49     256,923,577,521,058,878,088,611,477,224,235,621,321,607                                  506,876,294,100,502,275,687
51     12,589,255,298,531,8...,987,545,444,758,743 (44 digits)                                3,548,134,058,703,515,929,815
53     616,873,509,628,062,...,389,726,793,178,407 (45 digits)                               24,836,938,410,924,611,508,707
55     30,226,801,971,775,0...,096,612,865,741,943 (47 digits)                              173,858,568,876,472,280,560,953
57     1,481,113,296,616,97...,734,030,421,355,207 (49 digits)                            1,217,009,982,135,305,963,926,677
59     72,574,551,534,231,9...,967,490,646,405,143 (50 digits)                            8,519,069,874,947,141,747,486,745
61     3,556,153,025,177,36...,407,041,673,852,007 (52 digits)                           59,633,489,124,629,992,232,407,216
63     174,251,498,233,690,...,945,042,018,748,343 (54 digits)                          417,434,423,872,409,945,626,850,517
65     8,538,323,413,450,84...,307,058,918,668,807 (55 digits)                        2,922,040,967,106,869,619,387,953,625
67     418,377,847,259,091,...,045,887,014,771,543 (57 digits)                       20,454,286,769,748,087,335,715,675,381
69     20,500,514,515,695,4...,248,463,723,805,607 (59 digits)                      143,180,007,388,236,611,350,009,727,669
71     1,004,525,211,269,07...,174,722,466,474,743 (61 digits)                    1,002,260,051,717,656,279,450,068,093,686
73     49,221,735,352,184,8...,561,400,857,262,407 (62 digits)                    7,015,820,362,023,593,956,150,476,655,802
75     2,411,865,032,257,05...,508,642,005,857,943 (64 digits)                   49,110,742,534,165,157,693,053,336,590,618
85     681,292,175,541,205,...,256,581,907,552,807 (72 digits)              825,404,249,771,713,805,347,147,428,078,522,216
95     192,448,176,927,753,...,224,874,137,973,943 (81 digits)       13,872,569,225,913,193,926,469,506,823,715,722,892,042
100     3,234,476,509,624,75...,459,636,928,060,001 (85 digits)      1,798,465,042,647,41...,569,649,349,251,249 (43 digits)*
200    10,461,838,291,314,3...,534,637,456,120,001 (170 digits)      3,234,476,509,624,75...,459,636,928,060,001 (85 digits)*
300    33,838,570,200,749,1...,841,001,584,180,001 (254 digits)     5,817,092,933,824,34...,721,127,496,191,249 (127 digits)*
400    109,450,060,433,611,...,994,729,312,240,001 (339 digits)     10,461,838,291,314,3...,534,637,456,120,001 (170 digits)*
500    354,013,649,449,525,...,611,820,640,300,001 (423 digits)     18,815,250,448,759,0...,761,742,043,131,249 (212 digits)*
600    1,145,048,833,231,02...,308,275,568,360,001 (508 digits)     33,838,570,200,749,1...,841,001,584,180,001 (254 digits)*
700    3,703,633,553,458,98...,700,094,096,420,001 (592 digits)     60,857,485,599,217,6...,075,492,990,071,249 (296 digits)*
800    11,979,315,728,921,1...,403,276,224,480,001 (677 digits)     109,450,060,433,611,...,994,729,312,240,001 (339 digits)*
900    38,746,815,326,573,9...,033,821,952,540,001 (761 digits)     196,842,107,605,496,...,046,380,337,011,249 (381 digits)*
1000    125,325,663,996,571,...,207,731,280,600,001 (846 digits)     354,013,649,449,525,...,611,820,640,300,001 (423 digits)*
2000  15,706,522,056,181,6...,351,822,561,200,001 (1,691 digits)     125,325,663,996,571,...,207,731,280,600,001 (846 digits)*
3000  1,968,430,305,767,76...,432,273,841,800,001 (2,536 digits)   44,366,995,681,111,4...,787,731,920,900,001 (1,268 digits)*
4000  246,694,835,101,319,...,449,085,122,400,001 (3,381 digits)   15,706,522,056,181,6...,351,822,561,200,001 (1,691 digits)*
5000  30,917,194,013,597,6...,402,256,403,000,001 (4,226 digits)   5,560,323,193,268,32...,900,003,201,500,001 (2,113 digits)*
6000  3,874,717,868,664,96...,291,787,683,600,001 (5,071 digits)   1,968,430,305,767,76...,432,273,841,800,001 (2,536 digits)*
7000  485,601,589,689,818,...,117,678,964,200,001 (5,916 digits)   696,851,196,231,891,...,948,634,482,100,001 (2,958 digits)*
8000  60,858,341,665,667,3...,879,930,244,800,001 (6,761 digits)   246,694,835,101,319,...,449,085,122,400,001 (3,381 digits)*
9000  7,627,112,078,979,99...,578,541,525,400,001 (7,606 digits)   87,333,338,874,567,2...,933,625,762,700,001 (3,803 digits)*
```

(Note that pre-0.8.2 the "(NNN digits)" count includes commas)

## Prolog

Works with: SWI-Prolog version 8.5.9

```%%% -*- Prolog -*-
%%%
%%% The Rosetta Code integer square root task, for SWI Prolog.
%%%

%% pow4gtx/2 -- Find a power of 4 greater than X.
pow4gtx(X, Q) :- pow4gtx(X, 1, Q), !.
pow4gtx(X, A, Q) :- X < A, Q is A.
pow4gtx(X, A, Q) :- A1 is A * 4,
pow4gtx(X, A1, Q).

%% isqrt/2 -- Find integer square root.
%% isqrt/3 -- Find integer square root and remainder.
isqrt(X, R) :- isqrt(X, R, _).
isqrt(X, R, Z) :- pow4gtx(X, Q),
isqrt(X, Q, 0, X, R, Z).
isqrt(_, 1, R0, Z0, R, Z) :- R is R0,
Z is Z0.
isqrt(X, Q, R0, Z0, R, Z) :- Q1 is Q // 4,
T is Z0 - R0 - Q1,
(T >= 0
-> R1 is (R0 // 2) + Q1,
isqrt(X, Q1, R1, T, R, Z)
;  R1 is R0 // 2,
isqrt(X, Q1, R1, Z0, R, Z)).

roots(N) :- roots(0, N).
roots(I, N) :- isqrt(I, R),
write(R),
(I =:= N; write(" ")),
I1 is I + 1,
(N < I1, !; roots(I1, N)).

rootspow7(N) :- rootspow7(1, N).
rootspow7(I, N) :- Pow7 is 7**I,
isqrt(Pow7, R),
format("~t~D~2|~t~D~87|~t~D~131|~n",
[I, Pow7, R]),
I1 is I + 2,
(N < I1, !; rootspow7(I1, N)).

main :-
format("isqrt(i) for 0 <= i <= 65:~2n"),
roots(65),
format("~3n"),
format("isqrt(7**i) for 1 <= i <= 73, i odd:~2n"),
format("~t~s~2|~t~s~87|~t~s~131|~n",
["i", "7**i", "isqrt(7**i)"]),
format("-----------------------------------------------------------------------------------------------------------------------------------~n"),
rootspow7(73),
halt.

:- initialization(main).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Instructions for GNU Emacs--
%%% local variables:
%%% mode: prolog
%%% prolog-indent-width: 2
%%% end:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
Output:
```\$ swipl isqrt.pl
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                 isqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## Python

Works with: Python version 2.7
```def isqrt ( x ):
q = 1
while q <= x :
q *= 4
z,r = x,0
while q > 1 :
q  /= 4
t,r = z-r-q,r/2
if t >= 0 :
z,r = t,r+q
return r

print ' '.join( '%d'%isqrt( n ) for n in xrange( 66 ))
print '\n'.join( '{0:114,} = isqrt( 7^{1:3} )'.format( isqrt( 7**n ),n ) for n in range( 1,204,2 ))
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
2 = isqrt( 7^  1 )
18 = isqrt( 7^  3 )
129 = isqrt( 7^  5 )
907 = isqrt( 7^  7 )
6,352 = isqrt( 7^  9 )
44,467 = isqrt( 7^ 11 )
311,269 = isqrt( 7^ 13 )
2,178,889 = isqrt( 7^ 15 )
15,252,229 = isqrt( 7^ 17 )
106,765,608 = isqrt( 7^ 19 )
747,359,260 = isqrt( 7^ 21 )
5,231,514,822 = isqrt( 7^ 23 )
36,620,603,758 = isqrt( 7^ 25 )
256,344,226,312 = isqrt( 7^ 27 )
1,794,409,584,184 = isqrt( 7^ 29 )
12,560,867,089,291 = isqrt( 7^ 31 )
87,926,069,625,040 = isqrt( 7^ 33 )
615,482,487,375,282 = isqrt( 7^ 35 )
4,308,377,411,626,977 = isqrt( 7^ 37 )
30,158,641,881,388,842 = isqrt( 7^ 39 )
211,110,493,169,721,897 = isqrt( 7^ 41 )
1,477,773,452,188,053,281 = isqrt( 7^ 43 )
10,344,414,165,316,372,973 = isqrt( 7^ 45 )
72,410,899,157,214,610,812 = isqrt( 7^ 47 )
506,876,294,100,502,275,687 = isqrt( 7^ 49 )
3,548,134,058,703,515,929,815 = isqrt( 7^ 51 )
24,836,938,410,924,611,508,707 = isqrt( 7^ 53 )
173,858,568,876,472,280,560,953 = isqrt( 7^ 55 )
1,217,009,982,135,305,963,926,677 = isqrt( 7^ 57 )
8,519,069,874,947,141,747,486,745 = isqrt( 7^ 59 )
59,633,489,124,629,992,232,407,216 = isqrt( 7^ 61 )
417,434,423,872,409,945,626,850,517 = isqrt( 7^ 63 )
2,922,040,967,106,869,619,387,953,625 = isqrt( 7^ 65 )
20,454,286,769,748,087,335,715,675,381 = isqrt( 7^ 67 )
143,180,007,388,236,611,350,009,727,669 = isqrt( 7^ 69 )
1,002,260,051,717,656,279,450,068,093,686 = isqrt( 7^ 71 )
7,015,820,362,023,593,956,150,476,655,802 = isqrt( 7^ 73 )
49,110,742,534,165,157,693,053,336,590,618 = isqrt( 7^ 75 )
343,775,197,739,156,103,851,373,356,134,328 = isqrt( 7^ 77 )
2,406,426,384,174,092,726,959,613,492,940,298 = isqrt( 7^ 79 )
16,844,984,689,218,649,088,717,294,450,582,086 = isqrt( 7^ 81 )
117,914,892,824,530,543,621,021,061,154,074,602 = isqrt( 7^ 83 )
825,404,249,771,713,805,347,147,428,078,522,216 = isqrt( 7^ 85 )
5,777,829,748,401,996,637,430,031,996,549,655,515 = isqrt( 7^ 87 )
40,444,808,238,813,976,462,010,223,975,847,588,606 = isqrt( 7^ 89 )
283,113,657,671,697,835,234,071,567,830,933,120,245 = isqrt( 7^ 91 )
1,981,795,603,701,884,846,638,500,974,816,531,841,720 = isqrt( 7^ 93 )
13,872,569,225,913,193,926,469,506,823,715,722,892,042 = isqrt( 7^ 95 )
97,107,984,581,392,357,485,286,547,766,010,060,244,299 = isqrt( 7^ 97 )
679,755,892,069,746,502,397,005,834,362,070,421,710,095 = isqrt( 7^ 99 )
4,758,291,244,488,225,516,779,040,840,534,492,951,970,665 = isqrt( 7^101 )
33,308,038,711,417,578,617,453,285,883,741,450,663,794,661 = isqrt( 7^103 )
233,156,270,979,923,050,322,173,001,186,190,154,646,562,631 = isqrt( 7^105 )
1,632,093,896,859,461,352,255,211,008,303,331,082,525,938,421 = isqrt( 7^107 )
11,424,657,278,016,229,465,786,477,058,123,317,577,681,568,950 = isqrt( 7^109 )
79,972,600,946,113,606,260,505,339,406,863,223,043,770,982,651 = isqrt( 7^111 )
559,808,206,622,795,243,823,537,375,848,042,561,306,396,878,562 = isqrt( 7^113 )
3,918,657,446,359,566,706,764,761,630,936,297,929,144,778,149,940 = isqrt( 7^115 )
27,430,602,124,516,966,947,353,331,416,554,085,504,013,447,049,581 = isqrt( 7^117 )
192,014,214,871,618,768,631,473,319,915,878,598,528,094,129,347,071 = isqrt( 7^119 )
1,344,099,504,101,331,380,420,313,239,411,150,189,696,658,905,429,502 = isqrt( 7^121 )
9,408,696,528,709,319,662,942,192,675,878,051,327,876,612,338,006,515 = isqrt( 7^123 )
65,860,875,700,965,237,640,595,348,731,146,359,295,136,286,366,045,605 = isqrt( 7^125 )
461,026,129,906,756,663,484,167,441,118,024,515,065,954,004,562,319,241 = isqrt( 7^127 )
3,227,182,909,347,296,644,389,172,087,826,171,605,461,678,031,936,234,687 = isqrt( 7^129 )
22,590,280,365,431,076,510,724,204,614,783,201,238,231,746,223,553,642,811 = isqrt( 7^131 )
158,131,962,558,017,535,575,069,432,303,482,408,667,622,223,564,875,499,679 = isqrt( 7^133 )
1,106,923,737,906,122,749,025,486,026,124,376,860,673,355,564,954,128,497,756 = isqrt( 7^135 )
7,748,466,165,342,859,243,178,402,182,870,638,024,713,488,954,678,899,484,295 = isqrt( 7^137 )
54,239,263,157,400,014,702,248,815,280,094,466,172,994,422,682,752,296,390,067 = isqrt( 7^139 )
379,674,842,101,800,102,915,741,706,960,661,263,210,960,958,779,266,074,730,470 = isqrt( 7^141 )
2,657,723,894,712,600,720,410,191,948,724,628,842,476,726,711,454,862,523,113,293 = isqrt( 7^143 )
18,604,067,262,988,205,042,871,343,641,072,401,897,337,086,980,184,037,661,793,056 = isqrt( 7^145 )
130,228,470,840,917,435,300,099,405,487,506,813,281,359,608,861,288,263,632,551,397 = isqrt( 7^147 )
911,599,295,886,422,047,100,695,838,412,547,692,969,517,262,029,017,845,427,859,782 = isqrt( 7^149 )
6,381,195,071,204,954,329,704,870,868,887,833,850,786,620,834,203,124,917,995,018,479 = isqrt( 7^151 )
44,668,365,498,434,680,307,934,096,082,214,836,955,506,345,839,421,874,425,965,129,358 = isqrt( 7^153 )
312,678,558,489,042,762,155,538,672,575,503,858,688,544,420,875,953,120,981,755,905,510 = isqrt( 7^155 )
2,188,749,909,423,299,335,088,770,708,028,527,010,819,810,946,131,671,846,872,291,338,571 = isqrt( 7^157 )
15,321,249,365,963,095,345,621,394,956,199,689,075,738,676,622,921,702,928,106,039,370,003 = isqrt( 7^159 )
107,248,745,561,741,667,419,349,764,693,397,823,530,170,736,360,451,920,496,742,275,590,023 = isqrt( 7^161 )
750,741,218,932,191,671,935,448,352,853,784,764,711,195,154,523,163,443,477,195,929,130,162 = isqrt( 7^163 )
5,255,188,532,525,341,703,548,138,469,976,493,352,978,366,081,662,144,104,340,371,503,911,136 = isqrt( 7^165 )
36,786,319,727,677,391,924,836,969,289,835,453,470,848,562,571,635,008,730,382,600,527,377,954 = isqrt( 7^167 )
257,504,238,093,741,743,473,858,785,028,848,174,295,939,938,001,445,061,112,678,203,691,645,679 = isqrt( 7^169 )
1,802,529,666,656,192,204,317,011,495,201,937,220,071,579,566,010,115,427,788,747,425,841,519,758 = isqrt( 7^171 )
12,617,707,666,593,345,430,219,080,466,413,560,540,501,056,962,070,807,994,521,231,980,890,638,309 = isqrt( 7^173 )
88,323,953,666,153,418,011,533,563,264,894,923,783,507,398,734,495,655,961,648,623,866,234,468,168 = isqrt( 7^175 )
618,267,675,663,073,926,080,734,942,854,264,466,484,551,791,141,469,591,731,540,367,063,641,277,182 = isqrt( 7^177 )
4,327,873,729,641,517,482,565,144,599,979,851,265,391,862,537,990,287,142,120,782,569,445,488,940,274 = isqrt( 7^179 )
30,295,116,107,490,622,377,956,012,199,858,958,857,743,037,765,932,009,994,845,477,986,118,422,581,921 = isqrt( 7^181 )
212,065,812,752,434,356,645,692,085,399,012,712,004,201,264,361,524,069,963,918,345,902,828,958,073,452 = isqrt( 7^183 )
1,484,460,689,267,040,496,519,844,597,793,088,984,029,408,850,530,668,489,747,428,421,319,802,706,514,166 = isqrt( 7^185 )
10,391,224,824,869,283,475,638,912,184,551,622,888,205,861,953,714,679,428,231,998,949,238,618,945,599,162 = isqrt( 7^187 )
72,738,573,774,084,984,329,472,385,291,861,360,217,441,033,676,002,755,997,623,992,644,670,332,619,194,135 = isqrt( 7^189 )
509,170,016,418,594,890,306,306,697,043,029,521,522,087,235,732,019,291,983,367,948,512,692,328,334,358,945 = isqrt( 7^191 )
3,564,190,114,930,164,232,144,146,879,301,206,650,654,610,650,124,135,043,883,575,639,588,846,298,340,512,620 = isqrt( 7^193 )
24,949,330,804,511,149,625,009,028,155,108,446,554,582,274,550,868,945,307,185,029,477,121,924,088,383,588,341 = isqrt( 7^195 )
174,645,315,631,578,047,375,063,197,085,759,125,882,075,921,856,082,617,150,295,206,339,853,468,618,685,118,393 = isqrt( 7^197 )
1,222,517,209,421,046,331,625,442,379,600,313,881,174,531,452,992,578,320,052,066,444,378,974,280,330,795,828,756 = isqrt( 7^199 )
8,557,620,465,947,324,321,378,096,657,202,197,168,221,720,170,948,048,240,364,465,110,652,819,962,315,570,801,294 = isqrt( 7^201 )
59,903,343,261,631,270,249,646,676,600,415,380,177,552,041,196,636,337,682,551,255,774,569,739,736,208,995,609,059 = isqrt( 7^203 )
```

## Quackery

```  [ dup size 3 / times
[ char , swap
i 1+ -3 * stuff ]
dup 0 peek char , =
if [ behead drop ] ]  is +commas (   \$ --> \$   )

[ over size -
space swap of
swap join ]           is justify ( \$ n --> \$   )

[ 1
[ 2dup < not while
2 << again ]
0
[ over 1 > while
dip
[ 2 >>
2dup - ]
dup 1 >>
unrot -
dup 0 < iff drop
else
[ 2swap nip
rot over + ]
again ]
nip swap ]            is sqrt+   (   n --> n n )

( sqrt+ returns the integer square root and remainder )
( i.e. isqrt+ of 28 is 5 remainder 3 as (5^2)+3 = 28  )
( To make it task compliant change the last line to   )
( "nip nip ]             is sqrt+   (   n --> n   )"  )

66 times [ i^ sqrt+ drop echo sp ] cr cr

73 times
[ 7 i^ 1+ ** sqrt+ drop
number\$ +commas 41 justify
echo\$ cr
2 step ]```

Output:

```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

2
18
129
907
6,352
44,467
311,269
2,178,889
15,252,229
106,765,608
747,359,260
5,231,514,822
36,620,603,758
256,344,226,312
1,794,409,584,184
12,560,867,089,291
87,926,069,625,040
615,482,487,375,282
4,308,377,411,626,977
30,158,641,881,388,842
211,110,493,169,721,897
1,477,773,452,188,053,281
10,344,414,165,316,372,973
72,410,899,157,214,610,812
506,876,294,100,502,275,687
3,548,134,058,703,515,929,815
24,836,938,410,924,611,508,707
173,858,568,876,472,280,560,953
1,217,009,982,135,305,963,926,677
8,519,069,874,947,141,747,486,745
59,633,489,124,629,992,232,407,216
417,434,423,872,409,945,626,850,517
2,922,040,967,106,869,619,387,953,625
20,454,286,769,748,087,335,715,675,381
143,180,007,388,236,611,350,009,727,669
1,002,260,051,717,656,279,450,068,093,686
7,015,820,362,023,593,956,150,476,655,802
```

## Racket

```#lang racket

;; Integer Square Root (using Quadratic Residue)
(define (isqrt x)
(define q-init       ; power of 4 greater than x
(let loop ([acc 1])
(if (<= acc x) (loop (* acc 4)) acc)))

(define-values (z r q)
(let loop ([z x] [r 0] [q q-init])
(if (<= q 1)
(values z r q)
(let* ([q (/ q 4)]
[t (- z r q)]
[r (/ r 2)])
(if (>= t 0)
(loop t (+ r q) q)
(loop z r q))))))

r)

(define (format-with-commas str #:chunk-size [size 3])
(define len (string-length str))
(define len-mod (modulo len size))
(define chunks
(for/list ([i (in-range len-mod len size)])
(substring str i (+ i size))))
(string-join (if (= len-mod 0)
chunks
(cons (substring str 0 len-mod) chunks))
","))

(displayln "Isqrt of integers (0 -> 65):")
(for ([i 66])
(printf "~a " (isqrt i)))

(displayln "\n\nIsqrt of odd powers of 7 (7 -> 7^73):")
(for/fold ([num 7]) ([i (in-range 1 74 2)])
(printf "Isqrt(7^~a) = ~a\n"
i
(format-with-commas (number->string (isqrt num))))
(* num 49))
```
Output:
```Isqrt of integers (0 -> 65):
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Isqrt of odd powers of 7 (7 -> 7^73):
Isqrt(7^1) = 2
Isqrt(7^3) = 18
Isqrt(7^5) = 129
Isqrt(7^7) = 907
Isqrt(7^9) = 6,352
Isqrt(7^11) = 44,467
Isqrt(7^13) = 311,269
Isqrt(7^15) = 2,178,889
Isqrt(7^17) = 15,252,229
Isqrt(7^19) = 106,765,608
Isqrt(7^21) = 747,359,260
Isqrt(7^23) = 5,231,514,822
Isqrt(7^25) = 36,620,603,758
Isqrt(7^27) = 256,344,226,312
Isqrt(7^29) = 1,794,409,584,184
Isqrt(7^31) = 12,560,867,089,291
Isqrt(7^33) = 87,926,069,625,040
Isqrt(7^35) = 615,482,487,375,282
Isqrt(7^37) = 4,308,377,411,626,977
Isqrt(7^39) = 30,158,641,881,388,842
Isqrt(7^41) = 211,110,493,169,721,897
Isqrt(7^43) = 1,477,773,452,188,053,281
Isqrt(7^45) = 10,344,414,165,316,372,973
Isqrt(7^47) = 72,410,899,157,214,610,812
Isqrt(7^49) = 506,876,294,100,502,275,687
Isqrt(7^51) = 3,548,134,058,703,515,929,815
Isqrt(7^53) = 24,836,938,410,924,611,508,707
Isqrt(7^55) = 173,858,568,876,472,280,560,953
Isqrt(7^57) = 1,217,009,982,135,305,963,926,677
Isqrt(7^59) = 8,519,069,874,947,141,747,486,745
Isqrt(7^61) = 59,633,489,124,629,992,232,407,216
Isqrt(7^63) = 417,434,423,872,409,945,626,850,517
Isqrt(7^65) = 2,922,040,967,106,869,619,387,953,625
Isqrt(7^67) = 20,454,286,769,748,087,335,715,675,381
Isqrt(7^69) = 143,180,007,388,236,611,350,009,727,669
Isqrt(7^71) = 1,002,260,051,717,656,279,450,068,093,686
Isqrt(7^73) = 7,015,820,362,023,593,956,150,476,655,802
```

## Raku

There is a task Integer roots that covers a similar operation, with the caveat that it will calculate any nth root (including 2) not just square roots.

See the Integer roots Raku entry.

```use Lingua::EN::Numbers;

sub isqrt ( \x ) { my ( \$X, \$q, \$r, \$t ) =  x, 1, 0 ;
\$q +<= 2 while \$q ≤ \$X ;
while \$q > 1 {
\$q +>= 2; \$t = \$X - \$r - \$q; \$r +>= 1;
if \$t ≥ 0 { \$X = \$t; \$r += \$q }
}
\$r
}

say (^66)».&{ isqrt \$_ }.Str ;

(1, 3…73)».&{ "7**\$_: " ~ comma(isqrt 7**\$_) }».say
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
7**1: 2
7**3: 18
7**5: 129
7**7: 907
7**9: 6,352
7**11: 44,467
7**13: 311,269
7**15: 2,178,889
7**17: 15,252,229
7**19: 106,765,608
7**21: 747,359,260
7**23: 5,231,514,822
7**25: 36,620,603,758
7**27: 256,344,226,312
7**29: 1,794,409,584,184
7**31: 12,560,867,089,291
7**33: 87,926,069,625,040
7**35: 615,482,487,375,282
7**37: 4,308,377,411,626,977
7**39: 30,158,641,881,388,842
7**41: 211,110,493,169,721,897
7**43: 1,477,773,452,188,053,281
7**45: 10,344,414,165,316,372,973
7**47: 72,410,899,157,214,610,812
7**49: 506,876,294,100,502,275,687
7**51: 3,548,134,058,703,515,929,815
7**53: 24,836,938,410,924,611,508,707
7**55: 173,858,568,876,472,280,560,953
7**57: 1,217,009,982,135,305,963,926,677
7**59: 8,519,069,874,947,141,747,486,745
7**61: 59,633,489,124,629,992,232,407,216
7**63: 417,434,423,872,409,945,626,850,517
7**65: 2,922,040,967,106,869,619,387,953,625
7**67: 20,454,286,769,748,087,335,715,675,381
7**69: 143,180,007,388,236,611,350,009,727,669
7**71: 1,002,260,051,717,656,279,450,068,093,686
7**73: 7,015,820,362,023,593,956,150,476,655,802```

## REXX

A fair amount of code was included so that the output aligns correctly.

```/*REXX program computes and displays the Isqrt  (integer square root)  of some integers.*/
numeric digits 200                               /*insure 'nuff decimal digs for results*/
parse arg range power base .                     /*obtain optional arguments from the CL*/
if range=='' | range==","  then range= 0..65     /*Not specified?  Then use the default.*/
if power=='' | power==","  then power= 1..73     /* "      "         "   "   "     "    */
if base =='' | base ==","  then base =     7     /* "      "         "   "   "     "    */
parse var  range   rLO  '..'  rHI;     if rHI==''  then rHI= rLO      /*handle a range? */
parse var  power   pLO  '..'  pHI;     if pHI==''  then pHI= pLO      /*   "   "   "    */
\$=
do j=rLO  to rHI  while rHI>0        /*compute Isqrt for a range of integers*/
\$= \$ commas( Isqrt(j) )              /*append the Isqrt to a list for output*/
end   /*j*/
\$= strip(\$)                                      /*elide the leading blank in the list. */
say center(' Isqrt for numbers: '   rLO   " ──► "  rHI' ',  length(\$),  "─")
say strip(\$)                                     /*\$  has a leading blank for 1st number*/
say
z= base ** pHI                                   /*compute  max. exponentiation product.*/
Lp= max(30, length( commas(       z) ) )         /*length of "          "          "    */
Lr= max(20, length( commas( Isqrt(z) ) ) )       /* "     "    "  "   "  Isqrt of above.*/
say 'index'   center(base"**index", Lp)       center('Isqrt', Lr)        /*show a title.*/
say '─────'   copies("─",           Lp)       copies('─',     Lr)        /*  "  " header*/

do j=pLO  to pHI  by 2  while pHI>0;                              x= base ** j
say center(j, 5)  right( commas(x), Lp)      right( commas( Isqrt(x) ),  Lr)
end   /*j*/                          /* [↑]  show a bunch of powers & Isqrt.*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _;  do jc=length(_)-3  to 1  by -3; _=insert(',', _, jc); end;  return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
Isqrt: procedure; parse arg x                    /*obtain the only passed argument  X.  */
x= x % 1                                  /*convert possible real X to an integer*/     /* ◄■■■■■■■  optional. */
q= 1                                      /*initialize the  Q  variable to unity.*/
do until q>x      /*find a  Q  that is greater than  X.  */
q= q * 4          /*multiply   Q   by four.              */
end   /*until*/
r= 0                                      /*R:    will be the integer sqrt of X. */
do while q>1                    /*keep processing while  Q  is > than 1*/
q= q % 4                        /*divide  Q  by four  (no remainder).  */
t= x - r - q                    /*compute a temporary variable.        */
r= r % 2                        /*divide  R  by two   (no remainder).  */
if t >= 0  then do              /*if   T  is non─negative  ...         */
x= t            /*recompute the value of  X            */
r= r + q        /*    "      "    "    "  R            */
end
end   /*while*/
return r                                  /*return the integer square root of X. */
```
output   when using the default inputs:
```───────────────────────────────────────────────── Isqrt for numbers:  0  ──►  65 ──────────────────────────────────────────────────
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

index                                      7**index                                                        Isqrt
───── ────────────────────────────────────────────────────────────────────────────────── ─────────────────────────────────────────
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
```

## RPL

Because RPL can only handle unsigned integers, a light change has been made in the proposed algorithm,

Works with: Halcyon Calc version 4.2.7
RPL code Comment
``` ≪
#1
WHILE DUP2 ≥ REPEAT
SL SL END

#0
WHILE OVER #1 > REPEAT
SWAP SR SR SWAP
DUP2 +
SWAP SR SWAP
IF 4 PICK SWAP DUP2 ≥ THEN
- 4 ROLL DROP ROT ROT
OVER +
ELSE DROP2 END
END ROT ROT DROP2
≫
´ISQRT’ STO
```
```ISQRT ( #n -- #sqrt(n) )
q ◄── 1
perform while q <= x
q ◄── q * 4
z ◄── x
r ◄── 0
perform  while q > 1
q ◄── q ÷ 4
u ◄── r + q
r ◄── r ÷ 2
if z >= u  then do
z ◄── z - u
r ◄── r + q
else remove u and copy of z from stack
end perform, clean stack

```
Input:
```≪ { } 0 65 FOR n n R→B ISQRT B→R + NEXT ≫ EVAL
≪ {} #7 1 11 START DUP ISQRT ROT SWAP + SWAP 49 * NEXT DROP ≫ EVAL
```
Output:
```2: { 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 }
1: { # 2d # 18d # 129d # 907d # 6352d # 44467d # 311269d # 2178889d # 15252229d # 106765608d # 747359260d }
```

## Ruby

Ruby already has Integer.sqrt, which results in the integer square root of a positive integer. It can be re-implemented as follows:

```module Commatize
refine Integer do
def commatize
self.to_s.gsub( /(\d)(?=\d{3}+(?:\.|\$))(\d{3}\..*)?/, "\\1,\\2")
end
end
end

using Commatize
def isqrt(x)
q, r = 1, 0
while (q <= x) do q <<= 2 end
while (q > 1) do
q >>= 2; t = x-r-q; r >>= 1
if (t >= 0) then x, r = t, r+q end
end
r
end

puts (0..65).map{|n| isqrt(n) }.join(" ")

1.step(73, 2) do |n|
print "#{n}:\t"
puts isqrt(7**n).commatize
end
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
1:	2
3:	18
5:	129
7:	907
9:	6,352
11:	44,467
13:	311,269
15:	2,178,889
17:	15,252,229
19:	106,765,608
21:	747,359,260
23:	5,231,514,822
25:	36,620,603,758
27:	256,344,226,312
29:	1,794,409,584,184
31:	12,560,867,089,291
33:	87,926,069,625,040
35:	615,482,487,375,282
37:	4,308,377,411,626,977
39:	30,158,641,881,388,842
41:	211,110,493,169,721,897
43:	1,477,773,452,188,053,281
45:	10,344,414,165,316,372,973
47:	72,410,899,157,214,610,812
49:	506,876,294,100,502,275,687
51:	3,548,134,058,703,515,929,815
53:	24,836,938,410,924,611,508,707
55:	173,858,568,876,472,280,560,953
57:	1,217,009,982,135,305,963,926,677
59:	8,519,069,874,947,141,747,486,745
61:	59,633,489,124,629,992,232,407,216
63:	417,434,423,872,409,945,626,850,517
65:	2,922,040,967,106,869,619,387,953,625
67:	20,454,286,769,748,087,335,715,675,381
69:	143,180,007,388,236,611,350,009,727,669
71:	1,002,260,051,717,656,279,450,068,093,686
73:	7,015,820,362,023,593,956,150,476,655,802
```

## Rust

```use num::BigUint;
use num::CheckedSub;
use num_traits::{One, Zero};

fn isqrt(number: &BigUint) -> BigUint {
let mut q: BigUint = One::one();
while q <= *number {
q <<= &2;
}

let mut z = number.clone();
let mut result: BigUint = Zero::zero();

while q > One::one() {
q >>= &2;
let t = z.checked_sub(&result).and_then(|diff| diff.checked_sub(&q));
result >>= &1;

if let Some(t) = t {
z = t;
result += &q;
}
}

result
}

fn with_thousand_separator(s: &str) -> String {
let digits: Vec<_> = s.chars().rev().collect();
let chunks: Vec<_> = digits
.chunks(3)
.map(|chunk| chunk.iter().collect::<String>())
.collect();

chunks.join(",").chars().rev().collect::<String>()
}

fn main() {
println!("The integer square roots of integers from 0 to 65 are:");
(0_u32..=65).for_each(|n| print!("{} ", isqrt(&n.into())));

println!("\nThe integer square roots of odd powers of 7 from 7^1 up to 7^74 are:");
(1_u32..75).step_by(2).for_each(|exp| {
println!(
"7^{:>2}={:>83} ISQRT: {:>42} ",
exp,
with_thousand_separator(&BigUint::from(7_u8).pow(exp).to_string()),
with_thousand_separator(&isqrt(&BigUint::from(7_u8).pow(exp)).to_string())
)
});
}
```
Output:
```The integer square roots of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
The integer square roots of odd powers of 7 from 7^1 up to 7^74 are:
7^ 1=                                                                                  7 ISQRT:                                          2
7^ 3=                                                                                343 ISQRT:                                         18
7^ 5=                                                                             16,807 ISQRT:                                        129
7^ 7=                                                                            823,543 ISQRT:                                        907
7^ 9=                                                                         40,353,607 ISQRT:                                      6,352
7^11=                                                                      1,977,326,743 ISQRT:                                     44,467
7^13=                                                                     96,889,010,407 ISQRT:                                    311,269
7^15=                                                                  4,747,561,509,943 ISQRT:                                  2,178,889
7^17=                                                                232,630,513,987,207 ISQRT:                                 15,252,229
7^19=                                                             11,398,895,185,373,143 ISQRT:                                106,765,608
7^21=                                                            558,545,864,083,284,007 ISQRT:                                747,359,260
7^23=                                                         27,368,747,340,080,916,343 ISQRT:                              5,231,514,822
7^25=                                                      1,341,068,619,663,964,900,807 ISQRT:                             36,620,603,758
7^27=                                                     65,712,362,363,534,280,139,543 ISQRT:                            256,344,226,312
7^29=                                                  3,219,905,755,813,179,726,837,607 ISQRT:                          1,794,409,584,184
7^31=                                                157,775,382,034,845,806,615,042,743 ISQRT:                         12,560,867,089,291
7^33=                                              7,730,993,719,707,444,524,137,094,407 ISQRT:                         87,926,069,625,040
7^35=                                            378,818,692,265,664,781,682,717,625,943 ISQRT:                        615,482,487,375,282
7^37=                                         18,562,115,921,017,574,302,453,163,671,207 ISQRT:                      4,308,377,411,626,977
7^39=                                        909,543,680,129,861,140,820,205,019,889,143 ISQRT:                     30,158,641,881,388,842
7^41=                                     44,567,640,326,363,195,900,190,045,974,568,007 ISQRT:                    211,110,493,169,721,897
7^43=                                  2,183,814,375,991,796,599,109,312,252,753,832,343 ISQRT:                  1,477,773,452,188,053,281
7^45=                                107,006,904,423,598,033,356,356,300,384,937,784,807 ISQRT:                 10,344,414,165,316,372,973
7^47=                              5,243,338,316,756,303,634,461,458,718,861,951,455,543 ISQRT:                 72,410,899,157,214,610,812
7^49=                            256,923,577,521,058,878,088,611,477,224,235,621,321,607 ISQRT:                506,876,294,100,502,275,687
7^51=                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 ISQRT:              3,548,134,058,703,515,929,815
7^53=                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 ISQRT:             24,836,938,410,924,611,508,707
7^55=                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 ISQRT:            173,858,568,876,472,280,560,953
7^57=                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 ISQRT:          1,217,009,982,135,305,963,926,677
7^59=                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 ISQRT:          8,519,069,874,947,141,747,486,745
7^61=              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 ISQRT:         59,633,489,124,629,992,232,407,216
7^63=            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 ISQRT:        417,434,423,872,409,945,626,850,517
7^65=          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 ISQRT:      2,922,040,967,106,869,619,387,953,625
7^67=        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 ISQRT:     20,454,286,769,748,087,335,715,675,381
7^69=     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 ISQRT:    143,180,007,388,236,611,350,009,727,669
7^71=  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 ISQRT:  1,002,260,051,717,656,279,450,068,093,686
7^73= 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 ISQRT:  7,015,820,362,023,593,956,150,476,655,802
```

## S-BASIC

This follows the algorithm given in the task description. The q = q * 4 computation, however, will result in overflow (and an endless loop!) for large values of x.

```comment
return integer square root of n using quadratic
residue algorithm. WARNING: the function will fail
for x > 16,383.
end
function isqrt(x = integer) = integer
var q, r, t = integer
q = 1
while q <= x do
q = q * 4       rem overflow may occur here!
r = 0
while q > 1 do
begin
q = q / 4
t = x - r - q
r = r / 2
if t >= 0 then
begin
x = t
r = r + q
end
end
end = r

rem - Exercise the function

var n, pow7 = integer
print "Integer square root of first 65 numbers"
for n=1 to 65
print using "#####";isqrt(n);
next n
print
print "Integer square root of odd powers of 7"
print "  n    7^n   isqrt"
print "------------------"
for n=1 to 3 step 2
pow7 = 7^n
print using "###  ####  ####";n; pow7; isqrt(pow7)
next n

end
```

An alternate version of isqrt() that can handle the full range of S-BASIC integer values (well, almost: it will fail for 32,767) looks like this.

```function isqrt(x = integer) = integer
var x0, x1 = integer
x1 = x
repeat
begin
x0 = x1
x1 = (x0 + x / x0) / 2
end
until x1 >= x0
end = x0
```
Output:

The output for 7^5 will be shown only if the alternate version of the function is used.

```Integer square root of first 65 numbers
1    1    1    2    2    2    2    2    3    3    3    3    3    3    3    4
4    4    4    4    4    4    4    4    5    5    5    5    5    5    5    5
5    5    5    6    6    6    6    6    6    6    6    6    6    6    6    6
7    7    7    7    7    7    7    7    7    7    7    7    7    7    7    8
8
Integer square root of odd powers of 7
n    7^n   isqrt
------------------
1     7     2
3   343    18
5 16807   129
```

## Scheme

Works with: CHICKEN version 5.3.0
Library: r7rs
Library: format

Adapting this to any given R7RS Scheme is probably mainly a matter of changing how output is done.

```(import (scheme base))
(import (scheme write))
(import (format)) ;; Common Lisp formatting for CHICKEN Scheme.

(define (find-a-power-of-4-greater-than-x x)
(let loop ((q 1))
(if (< x q)
q
(loop (* 4 q)))))

(define (isqrt+remainder x)
(let loop ((q (find-a-power-of-4-greater-than-x x))
(z x)
(r 0))
(if (= q 1)
(values r z)
(let* ((q (truncate-quotient q 4))
(t (- z r q))
(r (truncate-quotient r 2)))
(if (negative? t)
(loop q z r)
(loop q t (+ r q)))))))

(define (isqrt x)
(let-values (((q r) (isqrt+remainder x)))
q))

(format #t "isqrt(i) for ~D <= i <= ~D:~2%" 0 65)
(do ((i 0 (+ i 1)))
((= i 65))
(format #t "~D " (isqrt i)))
(format #t "~D~3%" (isqrt 65))

(format #t "isqrt(7**i) for ~D <= i <= ~D, i odd:~2%" 1 73)
(format #t "~2@A ~84@A ~43@A~%" "i" "7**i" "sqrt(7**i)")
(format #t "~A~%" (make-string 131 #\-))
(do ((i 1 (+ i 2)))
((= i 75))
(let ((7**i (expt 7 i)))
(format #t "~2D ~84:D ~43:D~%" i 7**i (isqrt 7**i))))
```
Output:
```\$ csc -O3 -R r7rs isqrt.scm && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## SETL

```program isqrt;
loop for i in [1..65] do
if i mod 13=0 then print(); end if;
end loop;

print();
loop for p in [1, 3..73] do
sqrtp := isqrt(7 ** p);
print("sqrt(7^" + lpad(str p,2) + ") = " + lpad(str sqrtp, 32));
end loop;

proc isqrt(x);
q := 1;
loop while q<=x do
q *:= 4;
end loop;
z := x;
r := 0;
loop while q>1 do
q div:= 4;
t := z-r-q;
r div:= 2;
if t>=0 then
z := t;
r +:= q;
end if;
end loop;
return r;
end proc;
end program;```
Output:
```    1    1    1    2    2    2    2    2    3    3    3    3    3
3    3    4    4    4    4    4    4    4    4    4    5    5
5    5    5    5    5    5    5    5    5    6    6    6    6
6    6    6    6    6    6    6    6    6    7    7    7    7
7    7    7    7    7    7    7    7    7    7    7    8    8

sqrt(7^ 1) =                                2
sqrt(7^ 3) =                               18
sqrt(7^ 5) =                              129
sqrt(7^ 7) =                              907
sqrt(7^ 9) =                             6352
sqrt(7^11) =                            44467
sqrt(7^13) =                           311269
sqrt(7^15) =                          2178889
sqrt(7^17) =                         15252229
sqrt(7^19) =                        106765608
sqrt(7^21) =                        747359260
sqrt(7^23) =                       5231514822
sqrt(7^25) =                      36620603758
sqrt(7^27) =                     256344226312
sqrt(7^29) =                    1794409584184
sqrt(7^31) =                   12560867089291
sqrt(7^33) =                   87926069625040
sqrt(7^35) =                  615482487375282
sqrt(7^37) =                 4308377411626977
sqrt(7^39) =                30158641881388842
sqrt(7^41) =               211110493169721897
sqrt(7^43) =              1477773452188053281
sqrt(7^45) =             10344414165316372973
sqrt(7^47) =             72410899157214610812
sqrt(7^49) =            506876294100502275687
sqrt(7^51) =           3548134058703515929815
sqrt(7^53) =          24836938410924611508707
sqrt(7^55) =         173858568876472280560953
sqrt(7^57) =        1217009982135305963926677
sqrt(7^59) =        8519069874947141747486745
sqrt(7^61) =       59633489124629992232407216
sqrt(7^63) =      417434423872409945626850517
sqrt(7^65) =     2922040967106869619387953625
sqrt(7^67) =    20454286769748087335715675381
sqrt(7^69) =   143180007388236611350009727669
sqrt(7^71) =  1002260051717656279450068093686
sqrt(7^73) =  7015820362023593956150476655802```

## Seed7

Seed7 has integer sqrt() and bigInteger sqrt() functions. These functions could be used if an integer square root is needed. But this task does not allow using the language's built-in sqrt() function. Instead the quadratic residue algorithm for finding the integer square root must be used.

```\$ include "seed7_05.s7i";
include "bigint.s7i";

const func string: commatize (in bigInteger: bigNum) is func
result
var string: stri is "";
local
var integer: index is 0;
begin
stri := str(bigNum);
for index range length(stri) - 3 downto 1 step 3 do
stri := stri[.. index] & "," & stri[succ(index) ..];
end for;
end func;

const func bigInteger: isqrt (in bigInteger: x) is func
result
var bigInteger: r is 0_;
local
var bigInteger: q is 1_;
var bigInteger: z is 0_;
var bigInteger: t is 0_;
begin
while q <= x do
q *:= 4_;
end while;
z := x;
while q > 1_ do
q := q mdiv 4_;
t := z - r - q;
r := r mdiv 2_;
if t >= 0_ then
z := t;
r +:= q;
end if;
end while;
end func;

const proc: main is func
local
var integer: number is 0;
var bigInteger: pow7 is 7_;
begin
writeln("The integer square roots of integers from 0 to 65 are:");
for number range 0 to 65 do
write(isqrt(bigInteger(number)) <& " ");
end for;
writeln("\n\nThe integer square roots of powers of 7 from 7**1 up to 7**73 are:");
writeln("power                                    7 ** power                                                integer square root");
writeln("----- --------------------------------------------------------------------------------- -----------------------------------------");
for number range 1 to 73 step 2 do
pow7 *:= 49_;
end for;
end func;```
Output:
```The integer square roots of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

The integer square roots of powers of 7 from 7**1 up to 7**73 are:
power                                    7 ** power                                                integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
```

## Sidef

Built-in:

```var n = 1234
say n.isqrt
say n.iroot(2)
```

Explicit implementation for the integer k-th root of n:

```func rootint(n, k=2) {
return 0 if (n == 0)
var (s, v) = (n, k - 1)
loop {
var u = ((v*s + (n // s**v)) // k)
break if (u >= s)
s = u
}
s
}
```

Implementation of integer square root of n (using the quadratic residue algorithm):

```func isqrt(x) { var (q, r) = (1, 0); while (q <= x) { q <<= 2 }
while (q > 1) { q >>= 2; var t = x-r+q; r >>= 1
if (t >= 0) { (x, r) = (t, r+q) } } r }

say isqrt.map(0..65).join(' '); printf("\n")

for n in (1..73 `by` 2) {
printf("isqrt(7^%-2d): %42s\n", n, isqrt(7**n).commify) }
```
Output:
```0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7^1 ):                                          2
isqrt(7^3 ):                                         18
isqrt(7^5 ):                                        129
isqrt(7^7 ):                                        907
isqrt(7^9 ):                                      6,352
isqrt(7^11):                                     44,467
isqrt(7^13):                                    311,269
isqrt(7^15):                                  2,178,889
isqrt(7^17):                                 15,252,229
isqrt(7^19):                                106,765,608
isqrt(7^21):                                747,359,260
isqrt(7^23):                              5,231,514,822
isqrt(7^25):                             36,620,603,758
isqrt(7^27):                            256,344,226,312
isqrt(7^29):                          1,794,409,584,184
isqrt(7^31):                         12,560,867,089,291
isqrt(7^33):                         87,926,069,625,040
isqrt(7^35):                        615,482,487,375,282
isqrt(7^37):                      4,308,377,411,626,977
isqrt(7^39):                     30,158,641,881,388,842
isqrt(7^41):                    211,110,493,169,721,897
isqrt(7^43):                  1,477,773,452,188,053,281
isqrt(7^45):                 10,344,414,165,316,372,973
isqrt(7^47):                 72,410,899,157,214,610,812
isqrt(7^49):                506,876,294,100,502,275,687
isqrt(7^51):              3,548,134,058,703,515,929,815
isqrt(7^53):             24,836,938,410,924,611,508,707
isqrt(7^55):            173,858,568,876,472,280,560,953
isqrt(7^57):          1,217,009,982,135,305,963,926,677
isqrt(7^59):          8,519,069,874,947,141,747,486,745
isqrt(7^61):         59,633,489,124,629,992,232,407,216
isqrt(7^63):        417,434,423,872,409,945,626,850,517
isqrt(7^65):      2,922,040,967,106,869,619,387,953,625
isqrt(7^67):     20,454,286,769,748,087,335,715,675,381
isqrt(7^69):    143,180,007,388,236,611,350,009,727,669
isqrt(7^71):  1,002,260,051,717,656,279,450,068,093,686
isqrt(7^73):  7,015,820,362,023,593,956,150,476,655,802
```

## Standard ML

Translation of: Scheme
Translation of: OCaml
Works with: MLton

```(*

The Rosetta Code integer square root task, in Standard ML.

Compile with, for example:

mlton isqrt.sml

*)

val zero = IntInf.fromInt (0)
val one = IntInf.fromInt (1)
val seven = IntInf.fromInt (7)
val word1 = Word.fromInt (1)
val word2 = Word.fromInt (2)

fun
find_a_power_of_4_greater_than_x (x) =
let
fun
loop (q) =
if x < q then
q
else
loop (IntInf.<< (q, word2))
in
loop (one)
end;

fun
isqrt (x) =
let
fun
loop (q, z, r) =
if q = one then
r
else
let
val q = IntInf.~>> (q, word2)
val t = z - r - q
val r = IntInf.~>> (r, word1)
in
if t < zero then
loop (q, z, r)
else
loop (q, t, r + q)
end
in
loop (find_a_power_of_4_greater_than_x (x), x, zero)
end;

fun
insert_separators (s, sep) =
(* Insert separator characters (such as #",", #".", #" ") in a numeral
that is already in string form. *)
let
fun
loop (revchars, i, newchars) =
case (revchars, i) of
([], _) => newchars
| (revchars, 3) => loop (revchars, 0, sep :: newchars)
| (c :: tail, i) => loop (tail, i + 1, c :: newchars)
in
implode (loop (rev (explode s), 0, []))
end;

fun
commas (s) =
(* Insert commas in a numeral that is already in string form. *)
insert_separators (s, #",");

fun
main () =
let
val i = ref 0
in
print ("isqrt(i) for 0 <= i <= 65:\n\n");

i := 0;
while !i < 65 do (
print (IntInf.toString (isqrt (IntInf.fromInt (!i))));
print (" ");
i := !i + 1
);
print (IntInf.toString (isqrt (IntInf.fromInt (65))));
print ("\n\n\n");

print ("isqrt(7**i) for 1 <= i <= 73, i odd:\n\n");
print ("\n");

i := 1;
while !i <= 131 do (
print ("-");
i := !i + 1
);
print ("\n");

i := 1;
while !i <= 73 do (
let
val pow7 = IntInf.pow (seven, !i)
val root = isqrt (pow7)
in
print (pad_with_spaces 85 (commas (IntInf.toString pow7)));
print (pad_with_spaces 44 (commas (IntInf.toString root)));
print ("\n");
i := !i + 2
end
)
end;

main ();

(* local variables: *)
(* mode: sml *)
(* sml-indent-level: 2 *)
(* sml-indent-args: 2 *)
(* end: *)
```
Output:
```\$ mlton isqrt.sml && ./isqrt
isqrt(i) for 0 <= i <= 65:

0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7**i) for 1 <= i <= 73, i odd:

i                                                                                 7**i                                  sqrt(7**i)
-----------------------------------------------------------------------------------------------------------------------------------
1                                                                                    7                                           2
3                                                                                  343                                          18
5                                                                               16,807                                         129
7                                                                              823,543                                         907
9                                                                           40,353,607                                       6,352
11                                                                        1,977,326,743                                      44,467
13                                                                       96,889,010,407                                     311,269
15                                                                    4,747,561,509,943                                   2,178,889
17                                                                  232,630,513,987,207                                  15,252,229
19                                                               11,398,895,185,373,143                                 106,765,608
21                                                              558,545,864,083,284,007                                 747,359,260
23                                                           27,368,747,340,080,916,343                               5,231,514,822
25                                                        1,341,068,619,663,964,900,807                              36,620,603,758
27                                                       65,712,362,363,534,280,139,543                             256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                           1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                          12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                          87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                         615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                       4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                      30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                     211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                   1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                  10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                  72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607                 506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743               3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407              24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943             173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207           1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143           8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007          59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343         417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807       2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543      20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607     143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743   1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407   7,015,820,362,023,593,956,150,476,655,802```

## Swift

Translation of: C++

Requires the attaswift BigInt package.

```import BigInt

func integerSquareRoot<T: BinaryInteger>(_ num: T) -> T {
var x: T = num
var q: T = 1
while q <= x {
q <<= 2
}
var r: T = 0
while q > 1 {
q >>= 2
let t: T = x - r - q
r >>= 1
if t >= 0 {
x = t
r += q
}
}
return r
}

func pad(string: String, width: Int) -> String {
if string.count >= width {
return string
}
return String(repeating: " ", count: width - string.count) + string
}

func commatize<T: BinaryInteger>(_ num: T) -> String {
let string = String(num)
var result = String()
result.reserveCapacity(4 * string.count / 3)
var i = 0
for ch in string {
if i > 0 && i % 3 == string.count % 3 {
result += ","
}
result.append(ch)
i += 1
}
return result
}

print("Integer square root for numbers 0 to 65:")
for n in 0...65 {
print(integerSquareRoot(n), terminator: " ")
}

let powerWidth = 83
let isqrtWidth = 42
print("\n\nInteger square roots of odd powers of 7 from 1 to 73:")
print(" n |\(pad(string: "7 ^ n", width: powerWidth)) |\(pad(string: "isqrt(7 ^ n)", width: isqrtWidth))")
print(String(repeating: "-", count: powerWidth + isqrtWidth + 6))
var p: BigInt = 7
for n in stride(from: 1, through: 73, by: 2) {
let power = pad(string: commatize(p), width: powerWidth)
let isqrt = pad(string: commatize(integerSquareRoot(p)), width: isqrtWidth)
print("\(pad(string: String(n), width: 2)) |\(power) |\(isqrt)")
p *= 49
}
```
Output:
```Integer square root for numbers 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 73:
n |                                                                              7 ^ n |                              isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 |                                                                                  7 |                                         2
3 |                                                                                343 |                                        18
5 |                                                                             16,807 |                                       129
7 |                                                                            823,543 |                                       907
9 |                                                                         40,353,607 |                                     6,352
11 |                                                                      1,977,326,743 |                                    44,467
13 |                                                                     96,889,010,407 |                                   311,269
15 |                                                                  4,747,561,509,943 |                                 2,178,889
17 |                                                                232,630,513,987,207 |                                15,252,229
19 |                                                             11,398,895,185,373,143 |                               106,765,608
21 |                                                            558,545,864,083,284,007 |                               747,359,260
23 |                                                         27,368,747,340,080,916,343 |                             5,231,514,822
25 |                                                      1,341,068,619,663,964,900,807 |                            36,620,603,758
27 |                                                     65,712,362,363,534,280,139,543 |                           256,344,226,312
29 |                                                  3,219,905,755,813,179,726,837,607 |                         1,794,409,584,184
31 |                                                157,775,382,034,845,806,615,042,743 |                        12,560,867,089,291
33 |                                              7,730,993,719,707,444,524,137,094,407 |                        87,926,069,625,040
35 |                                            378,818,692,265,664,781,682,717,625,943 |                       615,482,487,375,282
37 |                                         18,562,115,921,017,574,302,453,163,671,207 |                     4,308,377,411,626,977
39 |                                        909,543,680,129,861,140,820,205,019,889,143 |                    30,158,641,881,388,842
41 |                                     44,567,640,326,363,195,900,190,045,974,568,007 |                   211,110,493,169,721,897
43 |                                  2,183,814,375,991,796,599,109,312,252,753,832,343 |                 1,477,773,452,188,053,281
45 |                                107,006,904,423,598,033,356,356,300,384,937,784,807 |                10,344,414,165,316,372,973
47 |                              5,243,338,316,756,303,634,461,458,718,861,951,455,543 |                72,410,899,157,214,610,812
49 |                            256,923,577,521,058,878,088,611,477,224,235,621,321,607 |               506,876,294,100,502,275,687
51 |                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 |             3,548,134,058,703,515,929,815
53 |                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 |            24,836,938,410,924,611,508,707
55 |                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 |           173,858,568,876,472,280,560,953
57 |                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 |         1,217,009,982,135,305,963,926,677
59 |                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 |         8,519,069,874,947,141,747,486,745
61 |              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 |        59,633,489,124,629,992,232,407,216
63 |            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 |       417,434,423,872,409,945,626,850,517
65 |          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 |     2,922,040,967,106,869,619,387,953,625
67 |        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 |    20,454,286,769,748,087,335,715,675,381
69 |     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 |   143,180,007,388,236,611,350,009,727,669
71 |  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
```

## Tiny BASIC

Works with: TinyBasic

Tiny BASIC does not support string formatting or concatenation, and is limited to integer arithmetic on numbers no greater than 32,767. The isqrt of 0-65 and the first two odd powers of 7 are shown in column format. The algorithm itself (the interesting part) begins on line 100.

```10 LET X = 0
20 GOSUB 100
30 PRINT R
40 LET X = X + 1
50 IF X < 66 THEN GOTO 20
70 PRINT "---"
71 LET X = 7
72 GOSUB 100
73 PRINT R
77 LET X = 343
78 GOSUB 100
79 PRINT R
90 END
100 REM integer square root function
110 LET Q = 1
120 IF Q > X THEN GOTO 150
130 LET Q = Q * 4
140 GOTO 120
150 LET Z = X
160 LET R = 0
170 IF Q <= 1 THEN RETURN
180 LET Q = Q / 4
190 LET T = Z - R - Q
200 LET R = R / 2
210 IF T < 0 THEN GOTO 170
220 LET Z = T
230 LET R = R + Q
240 GOTO 170
```

## UNIX Shell

Works with: Bourne Again SHell
Works with: Korn Shell
Works with: Zsh
```function isqrt {
typeset -i x
for x; do
typeset -i q=1
while (( q <= x )); do
(( q <<= 2 ))
if (( q <= 0 )); then
return 1
fi
done
typeset -i z=x
typeset -i r=0
typeset -i t
while (( q > 1 )); do
(( q >>= 2 ))
(( t = z - r - q ))
(( r >>= 1 ))
if (( t >= 0 )); then
(( z = t ))
(( r = r + q ))
fi
done
printf '%d\n' "\$r"
done
}

# demo
printf 'isqrt(n) for n from 0 to 65:\n'
for i in {1..4}; do
for n in {0..65}; do
case \$i in
1)
(( tens=n/10 ))
if (( tens )); then
printf '%2d' "\$tens"
else
printf '  '
fi
;;
2) printf '%2d' \$(( n%10 ));;
3) printf -- '--';;
4) printf '%2d' "\$(isqrt "\$n")";;
esac
done
printf '\n'
done
printf '\n'

printf 'isqrt(7ⁿ) for odd n up to the limit of integer precision:\n'
printf '%2s|%27sⁿ|%14sⁿ)\n' "n" "7" "isqrt(7"
for (( i=0;i<48; ++i )); do printf '-'; done; printf '\n'
for (( p=1; p<=73 && (n=7**p) > 0; p+=2)); do
if r=\$(isqrt \$n); then
printf "%2d|%'28d|%'16d\n" "\$p" "\$n" "\$r"
else
break
fi
done
```
Output:

The powers-of-7 table is limited by the built-in precision; on my system, both bash and zsh use signed 64-bit integers with a max value of 7²² < 9223372036854775807 < 7²³. Ksh uses signed 32-bit integers with a max value of 7¹¹ < 2147483647 < 7¹²; if I remove the typeset -i integer restriction, the code will work to a much larger power of 7, but at that point it's doing floating-point arithmetic, which is against the spirit of the task.

```isqrt(n) for n from 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
------------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7ⁿ) for odd n up to the limit of integer precision:
n|                          7ⁿ|       isqrt(7ⁿ)
------------------------------------------------
1|                           7|               2
3|                         343|              18
5|                      16,807|             129
7|                     823,543|             907
9|                  40,353,607|           6,352 # ksh stops here
11|               1,977,326,743|          44,467
13|              96,889,010,407|         311,269
15|           4,747,561,509,943|       2,178,889
17|         232,630,513,987,207|      15,252,229
19|      11,398,895,185,373,143|     106,765,608
21|     558,545,864,083,284,007|     747,359,260```

## Visual Basic .NET

Translation of: C#
```Imports System
Imports System.Console
Imports BI = System.Numerics.BigInteger

Module Module1
Function isqrt(ByVal x As BI) As BI
Dim t As BI, q As BI = 1, r As BI = 0
While q <= x : q <<= 2 : End While
While q > 1 : q >>= 2 : t = x - r - q : r >>= 1
If t >= 0 Then x = t : r += q
End While : Return r
End Function

Sub Main()
Const max As Integer = 73, smax As Integer = 65
Dim power_width As Integer = ((BI.Pow(7, max).ToString().Length \ 3) << 2) + 3,
isqrt_width As Integer = (power_width + 1) >> 1,
n as Integer
WriteLine("Integer square root for numbers 0 to {0}:", smax)
For n = 0 To smax : Write("{0} ", (n \ 10).ToString().Replace("0", " "))
Next : WriteLine()
For n = 0 To smax : Write("{0} ", n Mod 10) : Next : WriteLine()
WriteLine(New String("-"c, (smax << 1) + 1))
For n = 0 To smax : Write("{0} ", isqrt(n)) : Next
WriteLine(vbLf & vbLf & "Integer square roots of odd powers of 7 from 1 to {0}:", max)
Dim s As String = String.Format("[0,2] |[1,{0}:n0] |[2,{1}:n0]",
power_width, isqrt_width).Replace("[", "{").Replace("]", "}")
WriteLine(s, "n", "7 ^ n", "isqrt(7 ^ n)")
WriteLine(New String("-"c, power_width + isqrt_width + 6))
Dim p As BI = 7 : n = 1 : While n <= max
WriteLine(s, n, p, isqrt(p)) : n += 2 : p = p * 49
End While
End Sub
End Module
```
Output:
```Integer square root for numbers 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
-----------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

Integer square roots of odd powers of 7 from 1 to 73:
n |                                                                              7 ^ n |                              isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 |                                                                                  7 |                                         2
3 |                                                                                343 |                                        18
5 |                                                                             16,807 |                                       129
7 |                                                                            823,543 |                                       907
9 |                                                                         40,353,607 |                                     6,352
11 |                                                                      1,977,326,743 |                                    44,467
13 |                                                                     96,889,010,407 |                                   311,269
15 |                                                                  4,747,561,509,943 |                                 2,178,889
17 |                                                                232,630,513,987,207 |                                15,252,229
19 |                                                             11,398,895,185,373,143 |                               106,765,608
21 |                                                            558,545,864,083,284,007 |                               747,359,260
23 |                                                         27,368,747,340,080,916,343 |                             5,231,514,822
25 |                                                      1,341,068,619,663,964,900,807 |                            36,620,603,758
27 |                                                     65,712,362,363,534,280,139,543 |                           256,344,226,312
29 |                                                  3,219,905,755,813,179,726,837,607 |                         1,794,409,584,184
31 |                                                157,775,382,034,845,806,615,042,743 |                        12,560,867,089,291
33 |                                              7,730,993,719,707,444,524,137,094,407 |                        87,926,069,625,040
35 |                                            378,818,692,265,664,781,682,717,625,943 |                       615,482,487,375,282
37 |                                         18,562,115,921,017,574,302,453,163,671,207 |                     4,308,377,411,626,977
39 |                                        909,543,680,129,861,140,820,205,019,889,143 |                    30,158,641,881,388,842
41 |                                     44,567,640,326,363,195,900,190,045,974,568,007 |                   211,110,493,169,721,897
43 |                                  2,183,814,375,991,796,599,109,312,252,753,832,343 |                 1,477,773,452,188,053,281
45 |                                107,006,904,423,598,033,356,356,300,384,937,784,807 |                10,344,414,165,316,372,973
47 |                              5,243,338,316,756,303,634,461,458,718,861,951,455,543 |                72,410,899,157,214,610,812
49 |                            256,923,577,521,058,878,088,611,477,224,235,621,321,607 |               506,876,294,100,502,275,687
51 |                         12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 |             3,548,134,058,703,515,929,815
53 |                        616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 |            24,836,938,410,924,611,508,707
55 |                     30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 |           173,858,568,876,472,280,560,953
57 |                  1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 |         1,217,009,982,135,305,963,926,677
59 |                 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 |         8,519,069,874,947,141,747,486,745
61 |              3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 |        59,633,489,124,629,992,232,407,216
63 |            174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 |       417,434,423,872,409,945,626,850,517
65 |          8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 |     2,922,040,967,106,869,619,387,953,625
67 |        418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 |    20,454,286,769,748,087,335,715,675,381
69 |     20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 |   143,180,007,388,236,611,350,009,727,669
71 |  1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
```

## VTL-2

The ISQRT routine starts at line 2000. As VTL-2 only has unsigned 16-bit arithmetic, only the roots of 7^1, 7^3 and 7^5 are shown as 7^7 is too large.

```1000 X=0
1010 #=2000
1020 \$=32
1030 ?=R
1040 X=X+1
1050 #=X=33=0*1070
1060 ?=""
1070 #=X<66*1010
1080 P=1
1090 X=7
1100 #=2000
1110 ?=""
1120 ?="Root 7^";
1130 ?=P
1140 ?="(";
1150 ?=X
1160 ?=") = ";
1170 ?=R
1180 X=X*49
1190 P=P+2
1200 #=P<6*1100
1210 #=9999
2000 A=!
2010 Q=1
2020 #=X>Q=0*2050
2030 Q=Q*4
2040 #=2020
2050 Z=X
2060 R=0
2070 #=Q<2*A
2080 Q=Q/4
2090 T=Z-R-Q
2100 I=Z<(R+Q)
2110 R=R/2
2120 #=I*2070
2130 Z=T
2140 R=R+Q
2150 #=2070```
Output:
``` 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5
5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Root 7^1(7) = 2
Root 7^3(343) = 18
Root 7^5(16807) = 129
```

## Wren

Library: Wren-big
Library: Wren-fmt
```import "/big" for BigInt
import "/fmt" for Fmt

var isqrt = Fn.new { |x|
if (!(x is BigInt && x >= BigInt.zero)) {
Fiber.abort("Argument must be a non-negative big integer.")
}
var q = BigInt.one
while (q <= x) q = q * 4
var z = x
var r = BigInt.zero
while (q > BigInt.one) {
q = q >> 2
var t = z - r - q
r = r >> 1
if (t >= 0) {
z = t
r = r + q
}
}
return r
}

System.print("The integer square roots of integers from 0 to 65 are:")
for (i in 0..65) System.write("%(isqrt.call(BigInt.new(i))) ")
System.print()

System.print("\nThe integer square roots of powers of 7 from 7^1 up to 7^73 are:\n")
System.print("power                                    7 ^ power                                                 integer square root")
System.print("----- --------------------------------------------------------------------------------- -----------------------------------------")
var pow7 = BigInt.new(7)
var bi49 = BigInt.new(49)
var i = 1
while (i <= 73) {
Fmt.print("\$2d \$,84s \$,41s", i, pow7, isqrt.call(pow7))
pow7 = pow7 * bi49
i = i + 2
}```
Output:
```The integer square roots of integers from 0 to 65 are:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:

power                                    7 ^ power                                                 integer square root
----- --------------------------------------------------------------------------------- -----------------------------------------
1                                                                                    7                                         2
3                                                                                  343                                        18
5                                                                               16,807                                       129
7                                                                              823,543                                       907
9                                                                           40,353,607                                     6,352
11                                                                        1,977,326,743                                    44,467
13                                                                       96,889,010,407                                   311,269
15                                                                    4,747,561,509,943                                 2,178,889
17                                                                  232,630,513,987,207                                15,252,229
19                                                               11,398,895,185,373,143                               106,765,608
21                                                              558,545,864,083,284,007                               747,359,260
23                                                           27,368,747,340,080,916,343                             5,231,514,822
25                                                        1,341,068,619,663,964,900,807                            36,620,603,758
27                                                       65,712,362,363,534,280,139,543                           256,344,226,312
29                                                    3,219,905,755,813,179,726,837,607                         1,794,409,584,184
31                                                  157,775,382,034,845,806,615,042,743                        12,560,867,089,291
33                                                7,730,993,719,707,444,524,137,094,407                        87,926,069,625,040
35                                              378,818,692,265,664,781,682,717,625,943                       615,482,487,375,282
37                                           18,562,115,921,017,574,302,453,163,671,207                     4,308,377,411,626,977
39                                          909,543,680,129,861,140,820,205,019,889,143                    30,158,641,881,388,842
41                                       44,567,640,326,363,195,900,190,045,974,568,007                   211,110,493,169,721,897
43                                    2,183,814,375,991,796,599,109,312,252,753,832,343                 1,477,773,452,188,053,281
45                                  107,006,904,423,598,033,356,356,300,384,937,784,807                10,344,414,165,316,372,973
47                                5,243,338,316,756,303,634,461,458,718,861,951,455,543                72,410,899,157,214,610,812
49                              256,923,577,521,058,878,088,611,477,224,235,621,321,607               506,876,294,100,502,275,687
51                           12,589,255,298,531,885,026,341,962,383,987,545,444,758,743             3,548,134,058,703,515,929,815
53                          616,873,509,628,062,366,290,756,156,815,389,726,793,178,407            24,836,938,410,924,611,508,707
55                       30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943           173,858,568,876,472,280,560,953
57                    1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207         1,217,009,982,135,305,963,926,677
59                   72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143         8,519,069,874,947,141,747,486,745
61                3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007        59,633,489,124,629,992,232,407,216
63              174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343       417,434,423,872,409,945,626,850,517
65            8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807     2,922,040,967,106,869,619,387,953,625
67          418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543    20,454,286,769,748,087,335,715,675,381
69       20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607   143,180,007,388,236,611,350,009,727,669
71    1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686
73   49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
```

## Yabasic

```// Rosetta Code problem: https://rosettacode.org/wiki/Isqrt_(integer_square_root)_of_X
// by Jjuanhdez, 06/2022

print "Integer square root of first 65 numbers:"
for n = 1 to 65
print isqrt(n) using("##");
next n
print : print
print "Integer square root of odd powers of 7"
print "  n |                7^n |     isqrt "
print "----|--------------------|-----------"
for n = 1 to 21 step 2
pow7 = 7 ^ n
print n using("###"), " | ", left\$(str\$(pow7,"%20.1f"),18), " | ", left\$(str\$(isqrt(pow7),"%11.1f"),9)

next n
end

sub isqrt(x)
q = 1
while q <= x
q = q * 4
wend
r = 0
while q > 1
q = q / 4
t = x - r - q
r = r / 2
if t >= 0 then
x = t
r = r + q
end if
wend
return int(r)
end sub```