# 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

```BEGIN # find squares n where n + 1 is prime #
[]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```
Output:
``` 1 4 16 36 100 196 256 400 576 676
```

## ALGOL W

Using the difference of two squares to optimise the primality tests and the square loop (similar to the Phix and Applescript samples), though with the small number of values to test, that probably doesn't affect runtime much...

```begin % find squares n where n + 1 is prime                                  %

% returns true if n is prime, false otherwise, uses trial division       %
logical procedure isPrime ( integer value n ) ;
if      n < 3        then n = 2
else if n rem 3 = 0  then n = 3
else if not odd( n ) then false
else begin
logical prime;
integer f, f2, toNext;
prime  := true;
f      := 5;
f2     := 25;
toNext := 24;           % note: ( 2n + 1 )^2 - ( 2n - 1 )^2 = 8n %
while f2 <= n and prime do begin
prime  := n rem f not = 0;
f      := f + 2;
f2     := toNext;
toNext := toNext + 8
end while_f2_le_n_and_prime ;
prime
end isPrime ;

% other than 1, the numbers must be even                                 %
if isPrime( 2 % i.e.: ( 1 * 1 ) + 1 % ) then write( ( " 1" ) );

begin
integer i2, toNext;
toNext := i2 := 4;              % note: ( 2n + 2 )^2 - 2n^2 = 8n + 4 %
while i2 < 1000 do begin
if isPrime( i2 + 1 ) then writeon( i_w := 1, s_w := 0, " ", i2 );
toNext := toNext + 8;
i2     := i2 + toNext
end while_i2_lt_1000
end

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

## AppleScript

```on isPrime(n)
if (n < 4) then return (n > 1)
if ((n mod 2 is 0) or (n mod 3 is 0)) then return false
repeat with i from 5 to (n ^ 0.5) div 1 by 6
if ((n mod i is 0) or (n mod (i + 2) is 0)) then return false
end repeat

return true
end isPrime

set output to {}
if (isPrime(1 * 1 + 1)) then set end of output to 1 * 1
repeat with sqrt from 2 to (1000 ^ 0.5) by 2
set n to sqrt * sqrt
if (isPrime(n + 1)) then set end of output to n
end repeat

return output

```
Output:
```{1, 4, 16, 36, 100, 196, 256, 400, 576, 676}
```
Translation of: Phix

The first Phix solution's method of incrementing the square is fun, but not more efficient in AppleScript than the more straightforward method above. It can be optimised slightly by incrementing the d variable by 8 instead of incrementing by 4 and multiplying the result by 2. Also by incrementing the square + 1 each time instead of the square itself, so that 1 only has to be subtracted from the hits instead of being added to every square.

```on task()
set output to {1}
set nPlus1 to 5
repeat with d from 12 to (1000 ^ 0.5 div 0.25) by 8
if (isPrime(nPlus1)) then set end of output to nPlus1 - 1
set nPlus1 to nPlus1 + d
end repeat

return output
```

## Arturo

```1..31 | select 'x -> prime? 1 + x^2
| map 'x -> x^2
| print
```
Output:
`1 4 16 36 100 196 256 400 576 676`

## AutoHotkey

Translation of: FreeBASIC
```n := 0
while ((n2 := (n+=2)**2) < 1000)
if isPrime(n2+1)
result .= (result ? ", ":"" ) n2
MsgBox % result := 1 ", " result
return

isPrime(n, i:=2){
while (i < Sqrt(n)+1)
if !Mod(n, i++)
return False
return True
}
```
Output:
`1, 4, 16, 36, 100, 196, 256, 400, 576, 676`

## AWK

```# syntax: GAWK -f FIND_SQUARES_N_WHERE_N+1_IS_PRIME.AWK
BEGIN {
start = 1
stop = 999
n = 2
n2 = n^2
printf("1")
count++
while (n2 < stop) {
if (is_prime(n2+1)) {
printf(" %d",n2)
count++
}
n += 2
n2 = n^2
}
printf("\nFind squares %d-%d: %d\n",start,stop,count)
exit(0)
}
function is_prime(n,  d) {
d = 5
if (n < 2) { return(0) }
if (n % 2 == 0) { return(n == 2) }
if (n % 3 == 0) { return(n == 3) }
while (d*d <= n) {
if (n % d == 0) { return(0) }
d += 2
if (n % d == 0) { return(0) }
d += 4
}
return(1)
}
```
Output:
```1 4 16 36 100 196 256 400 576 676
Find squares 1-999: 10
```

## BASIC

```10 DEFINT A-Z: N=1000
20 DIM C(N)
30 FOR P=2 TO SQR(N)
40 IF NOT C(P) THEN FOR C=P*P TO N STEP P: C(C)=1=1: NEXT
50 NEXT
60 FOR I=2 TO N
70 IF C(I) THEN 100
80 X=I-1: R=SQR(X)
90 IF R*R=X THEN PRINT X;
100 NEXT
```
Output:
` 1  4  16  36  100  196  256  400  576  676`

## 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')
\$)```
Output:
`1 4 16 36 100 196 256 400 576 676`

## C

```#include <stdio.h>
#include <stdbool.h>
#include <math.h>

#define MAX 1000

void sieve(int n, bool *prime) {
prime[0] = prime[1] = false;
for (int i=2; i<=n; i++) prime[i] = true;
for (int p=2; p*p<=n; p++)
if (prime[p])
for (int c=p*p; c<=n; c+=p) prime[c] = false;
}

bool square(int n) {
int sq = sqrt(n);
return (sq * sq == n);
}

int main() {
bool prime[MAX + 1];
sieve(MAX, prime);
for (int i=2; i<=MAX; i++) if (prime[i]) {
int sq = i-1;
if (square(sq)) printf("%d ", sq);
}
printf("\n");
return 0;
}
```
Output:
`1 4 16 36 100 196 256 400 576 676`

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

## COBOL

```       IDENTIFICATION DIVISION.
PROGRAM-ID. SQUARE-PLUS-1-PRIME.

DATA DIVISION.
WORKING-STORAGE SECTION.
01 N                   PIC 999.
01 P                   PIC 9999 VALUE ZERO.
01 PRIMETEST.
03 DSOR             PIC 9999.
03 PRIME-FLAG       PIC X.
88 PRIME         VALUE '*'.
03 DIVTEST          PIC 9999V999.
03 FILLER           REDEFINES DIVTEST.
05 FILLER        PIC 9999.
05 FILLER        PIC 999.
88 DIVISIBLE  VALUE ZERO.

PROCEDURE DIVISION.
BEGIN.
PERFORM CHECK-N VARYING N FROM 1 BY 1
UNTIL P IS GREATER THAN 1000.
STOP RUN.

CHECK-N.
MULTIPLY N BY N GIVING P.
PERFORM CHECK-PRIME.
SUBTRACT 1 FROM P.
IF PRIME, DISPLAY P.

CHECK-PRIME.
IF P IS LESS THAN 2, MOVE SPACE TO PRIME-FLAG,
ELSE, MOVE '*' TO PRIME-FLAG.
PERFORM CHECK-DSOR VARYING DSOR FROM 2 BY 1
UNTIL NOT PRIME OR DSOR IS GREATER THAN N.

CHECK-DSOR.
DIVIDE P BY DSOR GIVING DIVTEST.
IF DIVISIBLE, MOVE SPACE TO PRIME-FLAG.
```
Output:
```0001
0004
0016
0036
0100
0196
0256
0400
0576
0676```

## Cowgol

```include "cowgol.coh";

const MAX := 1000;

sub isqrt(s: uint16): (x0: uint16) is
x0 := s>>1;
if x0 == 0 then
x0 := s;
else
loop
var x1: uint16 := (x0 + s/x0) >> 1;
if x1 >= x0 then return; end if;
x0 := x1;
end loop;
end if;
end sub;

var prime: uint8[MAX + 1];
MemSet(&prime[0], 1, @bytesof prime);

var p: uint16 := 2;
while p*p <= MAX loop
if prime[p] != 0 then
var c := p*p;
while c <= MAX loop
prime[c] := 0;
c := c + p;
end loop;
end if;
p := p + 1;
end loop;

var i: uint16 := 2;
while i <= MAX loop
if prime[i] != 0 then
var sq := i - 1;
var sqr := isqrt(sq);
if sqr*sqr == sq then
print_i16(sq);
print_nl();
end if;
end if;
i := i + 1;
end loop;```
Output:
```1
4
16
36
100
196
256
400
576
676```

## Delphi

Works with: Delphi version 6.0

```function IsPrime(N: integer): boolean;
{Fast, optimised prime test}
var I,Stop: integer;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (I + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;

procedure ShowPrimeSquares(Memo: TMemo);
var N,S2: integer;
begin
for N:= 1 to Trunc(sqrt(1000-1)) do
begin
S2:=N*N;
if IsPrime(S2+1) then Memo.Text:=Memo.Text+' '+IntToStr(S2);
end;
end;
```
Output:
``` 1 4 16 36 100 196 256 400 576 676
```

## EasyLang

```fastfunc isprim num .
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
n0 = 1
repeat
n = n0 * n0
until n >= 1000
if isprim (n + 1) = 1
write n & " "
.
n0 += 1
.```
Output:
```1 4 16 36 100 196 256 400 576 676
```

## F#

This task uses Extensible Prime Generator (F#)

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

## Fermat

```!!1;
i:=2;
i2:=4;
while i2<1000 do
if Isprime(i2+1) then !!i2 fi;
i:+2;
i2:=i^2;
od;```
Output:
```1
4
16
36
100
196
256
400
576
676

```

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

## Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"math"
"rcu"
)

func main() {
var squares []int
limit := int(math.Sqrt(1000))
i := 1
for i <= limit {
n := i * i
if rcu.IsPrime(n + 1) {
squares = append(squares, n)
}
if i == 1 {
i = 2
} else {
i += 2
}
}
fmt.Println("There are", len(squares), "square numbers 'n' where 'n+1' is prime, viz:")
fmt.Println(squares)
}
```
Output:
```There are 10 square numbers 'n' where 'n+1' is prime, viz:
[1 4 16 36 100 196 256 400 576 676]
```

## GW-BASIC

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

```

```module Squares
where

isPrime :: Int -> Bool
isPrime n
|n == 2 = True
|n == 1 = False
|otherwise = null \$ filter (\i -> mod n i == 0 ) [2 .. root]
where
root :: Int
root = floor \$ sqrt \$ fromIntegral n

isSquare :: Int -> Bool
isSquare n = theFloor * theFloor == n
where
theFloor :: Int
theFloor = floor \$ sqrt \$ fromIntegral n

solution :: [Int]
solution = [d | d <- [1..999] , isSquare d && isPrime ( d + 1 )]
```
Output:
```[1,4,16,36,100,196,256,400,576,676]
```

## J

```((<.=])@%:#+)@(i.&.(p:^:_1)-1:) 1000
```
Output:
`1 4 16 36 100 196 256 400 576 676`

## jq

Works with: jq

Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable definition of `is_prime` as used here.

```def squares_for_which_successor_is_prime:
(. // infinite) as \$limit
| {i:1, sq: 1}
| while( .sq < \$limit; .i += 1 | .sq = .i*.i)
| .sq
| select((. + 1)|is_prime) ;

1000 | squares_for_which_successor_is_prime```
Output:
```1
4
16
36
100
196
256
400
576
676
```

## Julia

```using Primes

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

foreach(p -> print(lpad(last(p), 5)), filter(isintegersquarebeforeprime, 1:1000))
```
Output:
`    1    4   16   36  100  196  256  400  576  676 `

```            NORMAL MODE IS INTEGER
BOOLEAN PRIME
DIMENSION PRIME(1000)

INTERNAL FUNCTION(S)
ENTRY TO ISQRT.
X0 = S/2
WHENEVER X0.E.0, FUNCTION RETURN S
FNDRT       X1 = (X0 + S/X0)/2
WHENEVER X1.GE.X0, FUNCTION RETURN X0
X0 = X1
TRANSFER TO FNDRT
END OF FUNCTION

THROUGH INIT, FOR P=2, 1, P.G.1000
INIT        PRIME(P) = 1B

THROUGH SIEVE, FOR P=2, 1, P*P.G.1000
THROUGH SIEVE, FOR C=P*P, P, C.G.1000
SIEVE       PRIME(C) = 0B

THROUGH TEST, FOR P=2, 1, P.G.1000
WHENEVER PRIME(P)
SQ = P-1
SQR = ISQRT.(SQ)
WHENEVER SQR*SQR.E.SQ
PRINT FORMAT FMT, SQ
END OF CONDITIONAL
END OF CONDITIONAL
TEST        CONTINUE

VECTOR VALUES FMT = \$I4*\$
END OF PROGRAM```
Output:
```   1
4
16
36
100
196
256
400
576
676```

## Mathematica / Wolfram Language

```Cases[Table[n^2, {n, 101}], _?(PrimeQ[# + 1] &)]
```
Output:
```
{1,4,16,36,100,196,256,400,576,676,1296,1600,2916,3136,4356,5476,7056,8100,8836}

```

## Modula-2

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

## Nim

```import std/strutils

func isPrime(n: Positive): bool =
if n < 2: return false
if (n and 1) == 0: return n == 2
var d = 3
while d * d <= n:
if n mod d == 0:
return false
inc d, 2
result = true

var list = @[1]
var n = 2
var n2 = 4
while n2 < 1000:
if isPrime(n2 + 1):
inc n, 2
n2 = n * n

echo list.join(" ")
```
Output:
```1 4 16 36 100 196 256 400 576 676
```

## OCaml

```let is_prime n =
let rec test x =
x * x > n || n mod x <> 0 && n mod (x + 2) <> 0 && test (x + 6)
in if n < 5 then n lor 1 = 3 else n land 1 <> 0 && n mod 3 <> 0 && test 5

let seq_squares =
let rec next n a () = Seq.Cons (n, next (n + a) (a + 2)) in
next 0 1

let () =
let cond n = is_prime (succ n) in
seq_squares |> Seq.take_while ((>) 1000) |> Seq.filter cond
|> Seq.iter (Printf.printf " %u") |> print_newline
```
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.

`for(n = 1, 1000, if(issquare(n)&&isprime(n+1),print(n)))`
Output:
```1
4
16
36
100
196
256
400
576

676```

## Perl

### Simple and Clear

```#!/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) };
```
Output:
```1 4 16 36 100 196 256 400 576 676
```

### More Than One Way

TMTOWTDI, right? So do it.

```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

# 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;
```
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," "))
```

## PL/M

Translation of: ALGOL W
Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

```100H: /* FIND SQUARES N WHERE N + ! IS PRIME                                */

/* CP/M BDOS SYSTEM CALL AND I/O ROUTINES                                */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C );  END;
PR\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S );  END;
PR\$NL:     PROCEDURE;   CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH ); END;
PR\$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH  */
DECLARE V ADDRESS, N\$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N\$STR );
N\$STR( W ) = '\$';
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR\$STRING( .N\$STR( W ) );
END PR\$NUMBER;

/* RETURNS TRUE IF N IS PRIME, FALSE OTHERWISE, USES TRIAL DIVISION      */
IS\$PRIME: PROCEDURE( N )BYTE;
DECLARE PRIME BYTE;
IF      N < 3       THEN PRIME = N = 2;
ELSE IF N MOD 3 = 0 THEN PRIME = N = 3;
ELSE IF N MOD 2 = 0 THEN PRIME = 0;
ELSE DO;
DECLARE ( F, F2, TO\$NEXT ) ADDRESS;
PRIME   = 1;
F       = 5;
F2      = 25;
TO\$NEXT = 24;            /* NOTE: ( 2N + 1 )^2 - ( 2N - 1 )^2 = 8N */
DO WHILE F2 <= N AND PRIME;
PRIME   = N MOD F <> 0;
F       = F + 2;
F2      = F2 + TO\$NEXT;
TO\$NEXT = TO\$NEXT + 8;
END;
END;
RETURN PRIME;
END IS\$PRIME;

/* OTHER THAN 1, THE NUMBERS MUST BE EVEN                                */
IF IS\$PRIME( 2 /* I.E.: ( 1 * 1 ) + 1 */ ) THEN DO;
CALL PR\$CHAR( ' ' );
CALL PR\$CHAR( '1' );
END;

DECLARE ( I2, TO\$NEXT ) ADDRESS;
TO\$NEXT, I2 = 4;                   /* NOTE: ( 2N + 2 )^2 - 2N^2 = 8N + 4 */
DO WHILE I2 < 1000;
IF IS\$PRIME( I2 + 1 ) THEN DO;
CALL PR\$CHAR( ' ' );
CALL PR\$NUMBER( I2 );
END;
TO\$NEXT = TO\$NEXT + 8;
I2      = I2 + TO\$NEXT;
END;

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

## PROMAL

Translation of: ALGOL W
```;;; Find squares n where n + 1 is prime
PROGRAM primesq
INCLUDE library

;;; returns TRUE(1) if p is prime, FALSE(0) otherwise
FUNC BYTE isPrime
ARG WORD n
WORD i
WORD f
WORD f2
WORD toNext
BYTE prime
BEGIN
IF        n < 3
prime = n = 2
ELSE IF   n % 3 = 0
prime = n = 3
ELSE IF   n % 2 = 0
prime = 0
ELSE
prime  = 1
f      = 5
f2     = 25
toNext = 24           ; note: ( 2n + 1 )^2 - ( 2n - 1 )^2 = 8n
WHILE f2 <= n AND prime
prime  = n % f <> 0
f      = f + 2
f2     = toNext
toNext = toNext + 8
RETURN prime
END

WORD i2
WORD toNext
BEGIN

IF isPrime( ( 1 * 1 ) + 1 )  ; 1 is the only possible odd number
OUTPUT " 1"

i2     = 4
toNext = 4                  ; note: ( 2n + 2 )^2 - 2n^2 = 8n + 4
WHILE i2 < 1000
IF isPrime( i2 + 1 )
OUTPUT " #W", i2
toNext = toNext + 8
i2     = i2 + toNext
END```
Output:
``` 1 4 16 36 100 196 256 400 576 676
```

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

## Quackery

`isprime` is defined at Primality by trial division#Quackery.

```  [] [] 0
[ 1+ dup 2 **
dup 1000 < while
1+ isprime if
[ dup dip join ]
again ]
2drop
witheach [ 2 ** join ]
echo```
Output:
`[ 1 4 16 36 100 196 256 400 576 676 ]`

## Racket

```#lang racket

(define (find-subprime-squares up-to)
(let rc ([curr-num 1]
[found '()])
(let ([n-sq (* curr-num curr-num)])
(cond [(>= n-sq up-to) (reverse found)]

(define (prime? n)
(let iter ([counter 2])
(cond [(eq? n 1) #f]
[(<= (expt counter 2) n)
(if (zero? (remainder n counter))
#f
[else #t])))

(find-subprime-squares 1000)
```
Output:
```'(1 4 16 36 100 196 256 400 576 676)
```

## 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...

```say ({\$++²}…^*>Ⅿ).grep: (*+1).is-prime
```
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...

```put (^Ⅿ).grep(*.is-prime).map(*-1).batch(20)».fmt("%3d").join: "\n"
```
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

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

## RPL

```≪ { }
1 1000 √ FOR j
IF j SQ 1 + ISPRIME? THEN j SQ + END
NEXT
```
Output:
```1: { 1 4 16 36 100 196 256 400 576 676 }
```

## Ruby

```require 'prime'

p (1..Integer.sqrt(1000)).filter_map{|n| sqr = n*n; sqr if (sqr+1).prime? }
```
Output:
```[1, 4, 16, 36, 100, 196, 256, 400, 576, 676]
```

## Rust

```use primes::is_prime ;

fn is_square( number : u64 ) -> bool {
let floor : u64 = (number as f64).sqrt( ).floor( ) as u64 ;
floor * floor == number
}

fn main() {
let solution : Vec<u64> = (1..1000).into_iter( ).
filter( | d | is_square( *d ) && is_prime( *d + 1 )).collect( ) ;
println!("{:?}" , solution);
}
```
Output:
```
[1, 4, 16, 36, 100, 196, 256, 400, 576, 676]

```

## Sidef

```1..1000.isqrt -> map { _**2 }.grep { is_prime(_+1) }.say
```
Output:
```[1, 4, 16, 36, 100, 196, 256, 400, 576, 676]
```

## Tiny BASIC

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

676```

## VTL-2

Translation of: TinyBASIC
```1000 ?=1
1010 N=2
1020 M=4
1030 J=M+1
1040 #=2000
1050 #=P=1=0*1080
1060 \$=32
1070 ?=M
1080 N=N+2
1090 M=N*N
1100 #=M<1000*1030
1110 #=9999
2000 R=!
2010 P=0
2020 I=3
2030 #=J/I*0+%=0*R
2040 I=I+2
2050 #=I*I<J*2030
2060 P=1
2070 #=R```
Output:
```1 4 16 36 100 196 256 400 576 676
```

## Wren

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

## 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, ^ )];
]```
Output:
```1 4 16 36 100 196 256 400 576 676
```