Isqrt (integer square root) of X: Difference between revisions
Not a robot (talk | contribs) Add Draco |
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=={{header|Delphi}}== |
=={{header|Delphi}}== |
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See [[#Pascal]]. |
See [[#Pascal]]. |
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=={{header|Draco}}== |
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Because all the intermediate values have to fit in a signed 32-bit integer, the largest power of 7 |
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for which the square root can be calculated is 7^10. |
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<lang draco>/* Integer square root using quadratic residue method */ |
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proc nonrec isqrt(ulong x) ulong: |
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ulong q, z, r; |
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long t; |
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q := 1; |
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while q <= x do q := q << 2 od; |
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z := x; |
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r := 0; |
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while q > 1 do |
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q := q >> 2; |
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t := z - r - q; |
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r := r >> 1; |
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if t >= 0 then |
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z := t; |
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r := r + q |
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fi |
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od; |
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r |
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corp |
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proc nonrec main() void: |
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byte x; |
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ulong pow7; |
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/* print isqrt(0) ... isqrt(65) */ |
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for x from 0 upto 65 do |
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write(isqrt(x):2); |
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if x % 11 = 10 then writeln() fi |
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od; |
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/* print isqrt(7^0) thru isqrt(7^10) */ |
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pow7 := 1; |
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for x from 0 upto 10 do |
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writeln("isqrt(7^", x:2, ") = ", isqrt(pow7):5); |
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pow7 := pow7 * 7 |
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od |
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corp</lang> |
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{{out}} |
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<pre> 0 1 1 1 2 2 2 2 2 3 3 |
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3 3 3 3 3 4 4 4 4 4 4 |
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4 4 4 5 5 5 5 5 5 5 5 |
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5 5 5 6 6 6 6 6 6 6 6 |
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6 6 6 6 6 7 7 7 7 7 7 |
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7 7 7 7 7 7 7 7 7 8 8 |
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isqrt(7^ 0) = 1 |
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isqrt(7^ 1) = 2 |
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isqrt(7^ 2) = 7 |
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isqrt(7^ 3) = 18 |
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isqrt(7^ 4) = 49 |
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isqrt(7^ 5) = 129 |
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isqrt(7^ 6) = 343 |
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isqrt(7^ 7) = 907 |
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isqrt(7^ 8) = 2401 |
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isqrt(7^ 9) = 6352 |
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isqrt(7^10) = 16807</pre> |
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=={{header|F_Sharp|F#}}== |
=={{header|F_Sharp|F#}}== |
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7,015,820,362,023,593,956,150,476,655,802 |
7,015,820,362,023,593,956,150,476,655,802 |
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</pre> |
</pre> |
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=={{header|Factor}}== |
=={{header|Factor}}== |
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The <code>isqrt</code> word is a straightforward translation of the pseudocode from the task description using lexical variables. |
The <code>isqrt</code> word is a straightforward translation of the pseudocode from the task description using lexical variables. |
Revision as of 15:25, 17 February 2022
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.
- Pseudo─code using quadratic residue
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
- Task
Compute and show all output here (on this page) for:
- 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.
- the
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.
- Related tasks
11l
<lang 11l>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</lang>
- 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
Ada
<lang Ada>with Ada.Text_Io; with Ada.Numerics.Big_Numbers.Big_Integers; with Ada.Strings.Fixed;
procedure Integer_Square_Root is
use Ada.Numerics.Big_Numbers.Big_Integers; use Ada.Text_Io;
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); use Ada.Strings.Fixed; 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;</lang>
- 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
Implements the task pseudo-code. <lang algol68>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 END; # pad left # # task test cases # 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</lang>
- 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-M
The approach here, while not the quadratic residue algorithm, works just fine. The output has been put into columnar form to avoid what would otherwise be an ugly mess on a typical 80 column display. <lang algol> BEGIN
% 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;
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 </lang> But for those for whom only the quadratic residue algorithm will do, just substitute this for the ISQRT() function in the previous example. (The output is identical.) But be warned: there is a bug lurking in the algorithm as presented in the task description. 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. <lang algol> 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; </lang>
- 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.
<lang applescript>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
-- Task code 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
on doTask()
-- 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
end doTask
doTask()</lang>
- Output:
<lang applescript>"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"</lang>
APL
Works in Dyalog APL <lang 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 }</lang>
- 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
<lang rebol>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]</lang>
- 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
C
Up to 64-bit limits with no big int library. <lang c>#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)); }
}</lang>
- 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++
<lang cpp>#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;
}</lang>
- 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#
<lang csharp>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)); }
}</lang>
- 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
<lang 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 array[char]$addl(out, ',') end end return(string$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</lang>
- 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
Cowgol
<lang 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;</lang>
- 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
D
<lang d>import std.bigint; import std.conv; import std.exception; import std.range; import std.regex; import std.stdio;
//Taken from the task http://rosettacode.org/wiki/Commatizing_numbers#D 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; }
}</lang>
- 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
Delphi
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. <lang draco>/* 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</lang>
- 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
F#
<lang fsharp> // 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))) </lang>
- 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.
<lang factor>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</lang>
- 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.
Código sacado de https://esolangs.org/wiki/Fish <lang Fish>1[:>:r:@@:@,\; ]~$\!?={:,2+/n</lang>
Forth
Only handles odd powers of 7 up to 7^21. <lang Forth>
- 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 ;
- task1
." Integer square roots from 0 to 65:" cr 66 0 do i isqrt . loop cr ;
- task2
." Integer square roots of 7^n" cr 7 11 0 do i 2* 1+ 2 .r 3 spaces dup isqrt ., cr 49 * loop ;
task1 cr task2 bye </lang> This version of the core word does not require locals. <lang>: 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 ;</lang>
- 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
<lang 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 .LT. 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 CHARACTER(2) :: HEADER_1 CHARACTER(83) :: HEADER_2 CHARACTER(83) :: HEADER_3
INTEGER(16) :: I
HEADER_1 = " n" HEADER_2 = "7^n" HEADER_3 = "isqrt(7^n)"
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(*,'(A,A,A,A,A)') HEADER_1, " | ", HEADER_2, " | ", HEADER_3 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</lang>
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. <lang freebasic> 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</lang>
- 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
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. <lang go>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.Add(t, z) t.Sub(t, r) t.Sub(t, q) r.Rsh(r, 1) if t.Cmp(zero) >= 0 { z.Set(t) r.Add(r, q) } } 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) }
}</lang>
- 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
Haskell
<lang haskell>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 <$> iterate (49 *) 7)</lang>
*Main> main [0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8] (1,2) (3,18) (5,129) (7,907) (9,6352) (11,44467) (13,311269) (15,2178889) (17,15252229) (19,106765608) (21,747359260) (23,5231514822) (25,36620603758) (27,256344226312) (29,1794409584184) (31,12560867089291) (33,87926069625040) (35,615482487375282) (37,4308377411626977) (39,30158641881388842) (41,211110493169721897) (43,1477773452188053281) (45,10344414165316372973) (47,72410899157214610812) (49,506876294100502275687) (51,3548134058703515929815) (53,24836938410924611508707) (55,173858568876472280560953) (57,1217009982135305963926677) (59,8519069874947141747486745) (61,59633489124629992232407216) (63,417434423872409945626850517) (65,2922040967106869619387953625) (67,20454286769748087335715675381) (69,143180007388236611350009727669) (71,1002260051717656279450068093686) (73,7015820362023593956150476655802)
J
Three implementations given. The floating point method is best for small square roots, Newton's method is fastest for extended integers. isqrt adapted from the page preamble. <lang J> isqrt_float=: <.@:%: isqrt_newton=: 9&$: :(x:@:<.@:-:@:(] + x:@:<.@:%)^:_&>~&:x:)
align=: (|.~ i.&' ')"1
comma=: (' ' -.~ [: }: [: , [: (|.) _3 (',' ,~ |.)\ |.)@":&>
While=: Template: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
) </lang>
(,. 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. 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
Java
<lang java>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; r = r.add(q); } } 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); } }
}</lang>
- 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
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.<lang jq># 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 task:
"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 | "\($i|lpad(2)) \($p|lpad(84)) \($p | isqrt | lpad(43))" )</lang>
- 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
Julia also has a built in isqrt() function which works on integer types, but the function integer_sqrt is shown for the task. <lang julia>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
println(lpad(i, 2), lpad(format(pow7^i, commas=true), 84), lpad(format(integer_sqrt(pow7^i), commas=true), 43))
end
</lang>
- 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
<lang scala>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 }
}</lang>
- 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
<lang lua>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</lang>
- 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
MAD
<lang MAD> 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 </lang>
- 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
<lang Mathematica>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])]</lang>
- 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
Nim
This Nim implementation provides an isqrt
function for signed integers and for big integers.
<lang Nim>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</lang>
- 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
Ol
<lang scheme> (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) </lang>
- 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
<lang Pascal> //************************************************// // // // Thanks for rvelthuis for BigIntegers library // // https://github.com/rvelthuis/DelphiBigNumbers // // // //************************************************//
program IsqrtTask;
{$APPTYPE CONSOLE}
{$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. </lang>
Perl
<lang Perl># 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";
}</lang>
- 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) mpz_add(r,r,q) 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)
Python
<lang python>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 ))</lang>
- 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
<lang 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 ]</lang>
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>
- 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))
</lang>
- 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.
Quadratic residue algorithm follows: <lang perl6>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</lang>
- 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. <lang rexx>/*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. */</lang>
- 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
Ruby
Ruby already has Integer.sqrt, which results in the integer square root of a positive integer. It can be re-implemented as follows: <lang ruby>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 </lang>
- 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
<lang 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()) ) });
} </lang>
- 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. <lang basic> 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 </lang> 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. <lang basic> 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 </lang>
- 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 n 7^n isqrt ------------------ 1 7 2 3 343 18
Seed7
<lang seed7>$ 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 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(sqrt(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 writeln(number lpad 2 <& commatize(pow7) lpad 85 <& commatize(sqrt(pow7)) lpad 42); pow7 := pow7 * 49_; end for; end func;</lang>
- 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: <lang ruby>var n = 1234 say n.isqrt say n.iroot(2)</lang>
Explicit implementation for the integer k-th root of n:
<lang ruby>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
}</lang>
Implementation of integer square root of n (using the quadratic residue algorithm): <lang ruby>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) }</lang>
- 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
Swift
Requires the attaswift BigInt package. <lang swift>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
}</lang>
- 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
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. <lang Tiny BASIC>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</lang>
UNIX Shell
<lang sh>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</lang>
- 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
<lang vbnet>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</lang>
- 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
Wren
<lang ecmascript>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
}</lang>
- 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
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