# Find squares n where n+1 is prime

Find squares n where n+1 is prime is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Find squares n where n+1 is prime and n<1.000

## ALGOL 68

<lang algol68>BEGIN # find squares n where n + 1 is prime #

```   PR read "primes.incl.a68" PR
[]BOOL prime = PRIMESIEVE 1 000; # construct a sieve of primes up to 1000 #
# find the squares 1 less than a prime (ignoring squares of non-integers) #
# other than 1, the numbers must be even                                  #
IF prime[ 2 # i.e.: ( 1 * 1 ) + 1 # ] THEN print( ( " 1" ) ) FI;
FOR i FROM 2 BY 2 TO UPB prime WHILE INT i2 = i * i;
i2 < UPB prime
DO
IF prime[ i2 + 1 ] THEN
print( ( " ", whole( i2, 0 ) ) )
FI
OD
```

END</lang>

Output:
``` 1 4 16 36 100 196 256 400 576 676
```

## BCPL

<lang bcpl>get "libhdr" manifest \$( MAX = 1000 \$)

let isqrt(s) = valof \$( let x0 = s>>1 and x1 = ?

```   if x0 = 0 resultis s
x1 := (x0 + s/x0)>>1
while x1<x0
\$(  x0 := x1
x1 := (x0 + s/x0)>>1
\$)
resultis x0
```

\$)

let sieve(prime, n) be \$( 0!prime := false

```   1!prime := false
for i = 2 to n do i!prime := true
for p = 2 to isqrt(n) if p!prime
\$(  let c = p*p
while c<n
\$(  c!prime := false
c := c + p
\$)
\$)
```

\$)

let square(n) = valof \$( let sq = isqrt(n)

```   resultis sq*sq = n
```

\$)

let start() be \$( let prime = vec MAX

```   sieve(prime, MAX)

for i=2 to MAX if i!prime
\$(  let sq = i-1
if square(sq) then writef("%N ",sq)
\$)
wrch('*N')
```

\$)</lang>

Output:
`1 4 16 36 100 196 256 400 576 676`

## CLU

<lang clu>isqrt = proc (s: int) returns (int)

```   x0: int := s/2
if x0=0 then return(s) end
x1: int := (x0 + s/x0)/2
while x1 < x0 do
x0 := x1
x1 := (x0 + s/x0)/2
end
return(x0)
```

end isqrt

sieve = proc (n: int) returns (array[int])

```   prime: array[bool] := array[bool]\$fill(2,n-1,true)
primes: array[int] := array[int]\$predict(1,isqrt(n))
for p: int in int\$from_to(2,isqrt(n)) do
if prime[p] then
for c: int in int\$from_to_by(p*p,n,p) do
prime[c] := false
end
end
end
for p: int in array[bool]\$indexes(prime) do
end
return(primes)
```

end sieve

is_square = proc (n: int) returns (bool)

```   return(isqrt(n) ** 2 = n)
```

end is_square

start_up = proc ()

```   po: stream := stream\$primary_output()
primes: array[int] := sieve(1000)

for prime: int in array[int]\$elements(primes) do
n: int := prime-1
if is_square(n) then stream\$puts(po, int\$unparse(n) || " ") end
end
```

end start_up</lang>

Output:
`1 4 16 36 100 196 256 400 576 676`

## F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Find squares n where n+1 is prime. Nigel Galloway: December 17th., 2021 seq{yield 1; for g in 2..2..30 do let n=g*g in if isPrime(n+1) then yield n}|>Seq.iter(printf "%d "); printfn "" </lang>

Output:
```1 4 16 36 100 196 256 400 576 676
```

## Fermat

<lang fermat>!!1; i:=2; i2:=4; while i2<1000 do

```   if Isprime(i2+1) then !!i2 fi;
i:+2;
i2:=i^2;
```

od;</lang>

Output:
```1
4
16
36
100
196
256
400
576
676

```

## FreeBASIC

<lang freebasic>function isprime(n as integer) as boolean

```   if n<0 then return isprime(-n)
if n<2 then return false
if n<4 then return true
dim as uinteger i=3
while i*i<n
if n mod i = 0 then return false
i+=2
wend
return true
```

end function

print 1;" "; dim as integer n=2, n2=4 while n2<1000

```   if isprime(1+n2) then print n2;"  ";
n+=2
n2=n^2
```

wend</lang>

Output:
` 1   4   16   36   100   196   256   400   576   676`

## GW-BASIC

<lang gwbasic>10 PRINT 1 20 N = 2 : N2 = 4 30 WHILE N2 < 1000 40 J = N2+1 50 GOSUB 110 60 IF PRIME = 1 THEN PRINT N2 70 N = N + 2 80 N2 = N*N 90 WEND 100 END 110 PRIME = 0 120 IF J < 2 THEN RETURN 130 PRIME = 1 140 IF J<4 THEN RETURN 150 I=5 160 WHILE I*I<J 170 IF J MOD I = 0 THEN PRIME = 0 : RETURN 180 I=I +2 190 WEND 200 RETURN</lang>

Output:
```1
4
16
36
100
196
256
400
576
676

```

## J

<lang j>((<.=])@%:#+)@(i.&.(p:^:_1)-1:) 1000</lang>

Output:
`1 4 16 36 100 196 256 400 576 676`

## Julia

<lang julia>using Primes

isintegersquarebeforeprime(n) = isqrt(n)^2 == n && isprime(n + 1)

foreach(p -> print(lpad(last(p), 5)), filter(isintegersquarebeforeprime, 1:1000))

</lang>
Output:
`    1    4   16   36  100  196  256  400  576  676 `

## Mathematica / Wolfram Language

<lang Mathematica>Cases[Table[n^2, {n, 101}], _?(PrimeQ[# + 1] &)]</lang>

Output:
```
{1,4,16,36,100,196,256,400,576,676,1296,1600,2916,3136,4356,5476,7056,8100,8836}

```

## Modula-2

<lang modula2>MODULE SquareAlmostPrime; FROM InOut IMPORT WriteCard, WriteLn; FROM MathLib IMPORT sqrt;

CONST Max = 1000;

VAR prime: ARRAY [0..Max] OF BOOLEAN;

```   i, sq: CARDINAL;

```

PROCEDURE Sieve;

```   VAR i, j, sqmax: CARDINAL;
```

BEGIN

```   sqmax := TRUNC(sqrt(FLOAT(Max)));
FOR i := 2 TO Max DO prime[i] := TRUE; END;
FOR i := 2 TO sqmax DO
IF prime[i] THEN
j := i * i;
WHILE j <= Max DO
prime[j] := FALSE;
j := j + i;
END;
END;
END;
```

END Sieve;

PROCEDURE isSquare(n: CARDINAL): BOOLEAN;

```   VAR sq: CARDINAL;
```

BEGIN

```   sq := TRUNC(sqrt(FLOAT(n)));
RETURN sq * sq = n;
```

END isSquare;

BEGIN

```   Sieve;
FOR i := 2 TO Max DO
IF prime[i] THEN
sq := i-1;
IF isSquare(sq) THEN
WriteCard(sq, 4);
WriteLn;
END;
END;
END;
```

END SquareAlmostPrime.</lang>

Output:
```   1
4
16
36
100
196
256
400
576
676```

## PARI/GP

This is not terribly efficient, but it does show off the issquare and isprime functions.

<lang parigp>for(n = 1, 1000, if(issquare(n)&&isprime(n+1),print(n)))</lang>

Output:
```1
4
16
36
100
196
256
400
576

676```

## Perl

### Simple and Clear

<lang perl>#!/usr/bin/perl use strict; # https://rosettacode.org/wiki/Find_squares_n_where_n%2B1_is_prime use warnings; use ntheory qw( primes is_square );

my @answer = grep is_square(\$_), map \$_ - 1, @{ primes(1000) }; print "@answer\n";</lang>

Output:
```1 4 16 36 100 196 256 400 576 676
```

### More Than One Way

TMTOWTDI, right? So do it. <lang perl>use strict; use warnings; use feature 'say'; use ntheory 'is_prime';

my \$a; is_prime \$_ and \$a = sqrt \$_-1 and \$a == int \$a and say \$_-1 for 1..1000; # backwards approach my \$b; do { say \$b**2 if is_prime 1 + ++\$b**2 } until \$b > int sqrt 1000; # do/until my \$c; while (++\$c < int sqrt 1000) { say \$c**2 if is_prime 1 + \$c**2 } # while/if say for map \$_**2, grep is_prime 1 + \$_**2, 1 .. int sqrt 1000; # for/map/grep for (1 .. int sqrt 1000) { say \$_**2 if is_prime 1 + \$_**2 } # for/if say \$_**2 for grep is_prime 1 + \$_**2, 1 .. int sqrt 1000; # for/grep is_prime 1 + \$_**2 and say \$_**2 for 1 .. int sqrt 1000; # and/for is_prime 1+\$_**2&&say\$_**2for 1..31; # and/for golf, FTW

1. or dispense with the module and find primes the slowest way possible

(1 x (1+\$_**2)) !~ /^(11+)\1+\$/ and say \$_**2 for 1 .. int sqrt 1000;</lang>

Output:

In all cases:

```1
4
16
36
100
196
256
400
576
676```

## Phix

```with javascript_semantics
sequence res = {1}
integer sq = 4, d = 2
while sq<1000 do
if is_prime(sq+1) then
res &= sq
end if
d += 4
sq += 2*d
end while
printf(1,"%V\n",{res})
```
Output:
```{1,4,16,36,100,196,256,400,576,676}
```

Alternative, same output, but 168 iterations/tests compared to just 16 by the above:

```with javascript_semantics
function sq(integer n) return integer(sqrt(n)) end function
pp(filter(sq_sub(get_primes_le(1000),1),sq))
```

Drop the filter to get the 168 (cheekily humorous) squares-of-integers-and-non-integers result of Raku (and format/arrange them identically):

```puts(1,join_by(apply(true,sprintf,{{"%3d"},sq_sub(get_primes_le(1000),1)}),1,20," "))
```

## Python

<lang python> limit = 1000 print("working...")

def isprime(n):

```   for i in range(2,int(n**0.5)+1):
if n%i==0:
return False
return True
```

def issquare(x): for n in range(1,x+1): if (x == n*n): return 1 return 0

for n in range(limit-1): if issquare(n) and isprime(n+1): print(n,end=" ")

print() print("done...") </lang>

Output:
```working...
1 4 16 36 100 196 256 400 576 676
done...
```

## Raku

Use up to to one thousand (1,000) rather than up to one (1.000) as otherwise it would be a pretty short list... <lang perl6>say ({\$++²}…^*>Ⅿ).grep: (*+1).is-prime</lang>

Output:
`(1 4 16 36 100 196 256 400 576 676)`

Although, technically, there is absolutely nothing in the task directions specifying that n needs to be the square of an integer. So, more accurately... <lang perl6>put (^Ⅿ).grep(*.is-prime).map(*-1).batch(20)».fmt("%3d").join: "\n"</lang>

Output:
```  1   2   4   6  10  12  16  18  22  28  30  36  40  42  46  52  58  60  66  70
72  78  82  88  96 100 102 106 108 112 126 130 136 138 148 150 156 162 166 172
178 180 190 192 196 198 210 222 226 228 232 238 240 250 256 262 268 270 276 280
282 292 306 310 312 316 330 336 346 348 352 358 366 372 378 382 388 396 400 408
418 420 430 432 438 442 448 456 460 462 466 478 486 490 498 502 508 520 522 540
546 556 562 568 570 576 586 592 598 600 606 612 616 618 630 640 642 646 652 658
660 672 676 682 690 700 708 718 726 732 738 742 750 756 760 768 772 786 796 808
810 820 822 826 828 838 852 856 858 862 876 880 882 886 906 910 918 928 936 940
946 952 966 970 976 982 990 996```

## Ring

<lang ring> load "stdlib.ring" row = 0 limit = 1000 see "working..." + nl

for n = 1 to limit-1

```   if issquare(n) and isprime(n+1)
row++
see "" + n +nl
ok
```

next

see "Found " + row + " numbers" + nl see "done..." + nl

func issquare(x)

```    for n = 1 to sqrt(x)
if x = pow(n,2)
return 1
ok
next
return 0
```

</lang>

Output:
```working...
1
4
16
36
100
196
256
400
576
676
Found 10 numbers
done...
```

## Tiny BASIC

<lang tinybasic> PRINT 1

```     LET N = 2
LET M = 4
10 LET J = M + 1
GOSUB 20
IF P = 1 THEN PRINT M
LET N = N + 2
LET M = N*N
IF M < 1000 THEN GOTO 10
END
20 LET P = 0
LET I = 3
30 IF (J/I)*I = J THEN RETURN
LET I = I + 2
IF I*I < J THEN GOTO 30
LET P = 1
RETURN</lang>
```
Output:
```1
4
16
36
100
196
256
400
576

676```

## Wren

Library: Wren-math

<lang ecmascript>import "./math" for Int

var squares = [] var limit = 1000.sqrt.floor var i = 1 while (i <= limit) {

```   var n = i * i
i = (i == 1) ? 2 : i + 2
```

} System.print("There are %(squares.count) square numbers 'n' where 'n+1' is prime, viz:") System.print(squares)</lang>

Output:
```There are 10 square numbers 'n' where 'n+1' is prime, viz:
[1, 4, 16, 36, 100, 196, 256, 400, 576, 676]
```

## XPL0

<lang XPL0>func IsPrime(N); \Return 'true' if N is prime int N, I; [if N <= 2 then return N = 2; if (N&1) = 0 then \even >2\ return false; for I:= 3 to sqrt(N) do

```   [if rem(N/I) = 0 then return false;
I:= I+1;
];
```

return true; ]; \IsPrime

int N; [for N:= 1 to sqrt(1000-1) do

```   if IsPrime(N*N+1) then
[IntOut(0, N*N);  ChOut(0, ^ )];
```

]</lang>

Output:
```1 4 16 36 100 196 256 400 576 676
```