# Find adjacent primes which differ by a square integer

Find adjacent primes which differ by a square integer 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 adjacent primes under 1,000,000 whose difference (> 36) is a square integer.

## 11l

```F primes_upto(limit)
V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
L(n) 0 .< Int(limit ^ 0.5 + 1.5)
I is_prime[n]
L(i) (n * n .. limit).step(n)
is_prime[i] = 0B
R enumerate(is_prime).filter((i, prime) -> prime).map((i, prime) -> i)

V primes = primes_upto(1'000'000)

F is_square(x)
R Int(sqrt(x)) ^ 2 == x

L(n) 2 .< primes.len
V pr1 = primes[n]
V pr2 = primes[n - 1]
V diff = pr1 - pr2
I (is_square(diff) & diff > 36)
print(pr1‘ ’pr2‘ diff = ’diff)```
Output:
```89753 89689 diff = 64
107441 107377 diff = 64
288647 288583 diff = 64
368021 367957 diff = 64
381167 381103 diff = 64
396833 396733 diff = 100
400823 400759 diff = 64
445427 445363 diff = 64
623171 623107 diff = 64
625763 625699 diff = 64
637067 637003 diff = 64
710777 710713 diff = 64
725273 725209 diff = 64
779477 779413 diff = 64
801947 801883 diff = 64
803813 803749 diff = 64
821741 821677 diff = 64
832583 832519 diff = 64
838349 838249 diff = 100
844841 844777 diff = 64
883871 883807 diff = 64
912167 912103 diff = 64
919511 919447 diff = 64
954827 954763 diff = 64
981887 981823 diff = 64
997877 997813 diff = 64
```

## ALGOL 68

```BEGIN # find a adjacent primes where the primes differ by a square > 36 #
INT min diff  = 37;
INT max prime = 1 000 000;
# form a list of primes to max prime #
[]INT prime = EXTRACTPRIMESUPTO max prime FROMPRIMESIEVE PRIMESIEVE max prime;
# construct a table of squares, we will need at most the square root of max prime #
# but in reality much less than that - assume 1000 will be enough #
[ 1 : 1000 ]BOOL is square;
FOR i TO UPB is square DO is square[ i ] := FALSE OD;
FOR i WHILE INT i2 = i * i;
i2 <= UPB is square
DO
is square[ i2 ] := TRUE
OD;
# find the primes #
FOR p TO UPB prime - 1 DO
INT q    = p + 1;
INT diff = prime[ q ] - prime[ p ];
IF diff > min diff AND is square[ diff ] THEN
print( ( whole( prime[ q ], -6 ), " - ", whole( prime[ p ], -6 ), " = ", whole( diff, 0 ), newline ) )
FI
OD
END```
Output:
``` 89753 -  89689 = 64
107441 - 107377 = 64
288647 - 288583 = 64
368021 - 367957 = 64
381167 - 381103 = 64
396833 - 396733 = 100
400823 - 400759 = 64
445427 - 445363 = 64
623171 - 623107 = 64
625763 - 625699 = 64
637067 - 637003 = 64
710777 - 710713 = 64
725273 - 725209 = 64
779477 - 779413 = 64
801947 - 801883 = 64
803813 - 803749 = 64
821741 - 821677 = 64
832583 - 832519 = 64
838349 - 838249 = 100
844841 - 844777 = 64
883871 - 883807 = 64
912167 - 912103 = 64
919511 - 919447 = 64
954827 - 954763 = 64
981887 - 981823 = 64
997877 - 997813 = 64
```

## AWK

```# syntax: GAWK -f FIND_ADJACENTS_PRIMES_WHICH_DIFFERENCE_IS_SQUARE_INTEGER.AWK
# converted from FreeBASIC
BEGIN {
start = i = 3
stop =  999999
while (j <= stop) {
j = next_prime(i)
if (j-i > 36 && is_square(j-i)) {
printf("%9d %9d %9d\n",i,j,j-i)
count++
}
i = j
}
printf("Adjacent primes which difference is square integer (>36) %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)
}
function is_square(n) {
return (int(sqrt(n))^2 == n)
}
function next_prime(n,  q) { # finds next prime after n
if (n == 0) { return(2) }
if (n < 3) { return(++n) }
q = n + 2
while (!is_prime(q)) {
q += 2
}
return(q)
}
```
Output:
```    89689     89753        64
107377    107441        64
288583    288647        64
367957    368021        64
381103    381167        64
396733    396833       100
400759    400823        64
445363    445427        64
623107    623171        64
625699    625763        64
637003    637067        64
710713    710777        64
725209    725273        64
779413    779477        64
801883    801947        64
803749    803813        64
821677    821741        64
832519    832583        64
838249    838349       100
844777    844841        64
883807    883871        64
912103    912167        64
919447    919511        64
954763    954827        64
981823    981887        64
997813    997877        64
Adjacent primes which difference is square integer (>36) 3-999999: 26
```

## C

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

int isprime( int p ) {
int i;
if(p==2) return 1;
if(!(p%2)) return 0;
for(i=3; i*i<=p; i+=2) {
if(!(p%i)) return 0;
}
return 1;
}

int nextprime( int p ) {
int i=0;
if(p==0) return 2;
if(p<3) return p+1;
while(!isprime(++i + p));
return i+p;
}

int issquare( int p ) {
int i;
for(i=0;i*i<p;i++);
return i*i==p;
}

int main(void) {
int i=3, j=2;
for(i=3;j<=1000000;i=j) {
j=nextprime(i);
if(j-i>36&&issquare(j-i)) printf( "%d %d %d\n", i, j, j-i );
}
return 0;
}
```

## CLU

```% Integer square root
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

% See if a number is square
% Note that all squares are 0, 1, 4, or 9 mod 16.
is_square = proc (n: int) returns (bool)
d: int := n//16
if d=0 cor d=1 cor d=4 cor d=9 then
return(n = isqrt(n)**2)
else
return(false)
end
end is_square

% Find all primes up to a given number
sieve = proc (top: int) returns (array[int])
prime: array[bool] := array[bool]\$fill(2,top-1,true)
for p: int in int\$from_to(2,isqrt(top)) do
if prime[p] then
for c: int in int\$from_to_by(p*p,top,p) do
prime[c] := false
end
end
end
list: array[int] := array[int]\$predict(1,isqrt(top))
for p: int in int\$from_to(2,top) do
end
return(list)
end sieve

start_up = proc ()
MAX = 1000000
DIFF = 36

po: stream := stream\$primary_output()
primes: array[int] := sieve(MAX)
for i: int in int\$from_to(array[int]\$low(primes)+1,
array[int]\$high(primes)) do
d: int := primes[i] - primes[i-1]
if d>DIFF cand is_square(d) then
stream\$putright(po, int\$unparse(primes[i]), 6)
stream\$puts(po, " - ")
stream\$putright(po, int\$unparse(primes[i-1]), 6)
stream\$puts(po, " = ")
stream\$putright(po, int\$unparse(d), 4)
stream\$puts(po, " = ")
stream\$putright(po, int\$unparse(isqrt(d)), 4)
stream\$putl(po, "^2")
end
end
end start_up```
Output:
``` 89753 -  89689 =   64 =    8^2
107441 - 107377 =   64 =    8^2
288647 - 288583 =   64 =    8^2
368021 - 367957 =   64 =    8^2
381167 - 381103 =   64 =    8^2
396833 - 396733 =  100 =   10^2
400823 - 400759 =   64 =    8^2
445427 - 445363 =   64 =    8^2
623171 - 623107 =   64 =    8^2
625763 - 625699 =   64 =    8^2
637067 - 637003 =   64 =    8^2
710777 - 710713 =   64 =    8^2
725273 - 725209 =   64 =    8^2
779477 - 779413 =   64 =    8^2
801947 - 801883 =   64 =    8^2
803813 - 803749 =   64 =    8^2
821741 - 821677 =   64 =    8^2
832583 - 832519 =   64 =    8^2
838349 - 838249 =  100 =   10^2
844841 - 844777 =   64 =    8^2
883871 - 883807 =   64 =    8^2
912167 - 912103 =   64 =    8^2
919511 - 919447 =   64 =    8^2
954827 - 954763 =   64 =    8^2
981887 - 981823 =   64 =    8^2
997877 - 997813 =   64 =    8^2```

## F#

This task uses Extensible Prime Generator (F#)

```// Find adjacents primes which difference is square integer . Nigel Galloway: November 23rd., 2021
primes32()|>Seq.takeWhile((>)1000000)|>Seq.pairwise|>Seq.filter(fun(n,g)->let n=g-n in let g=(float>>sqrt>>int)n in g>6 && n=g*g)|>Seq.iter(printfn "%A")
```
Output:
```(89689, 89753)
(107377, 107441)
(288583, 288647)
(367957, 368021)
(381103, 381167)
(396733, 396833)
(400759, 400823)
(445363, 445427)
(623107, 623171)
(625699, 625763)
(637003, 637067)
(710713, 710777)
(725209, 725273)
(779413, 779477)
(801883, 801947)
(803749, 803813)
(821677, 821741)
(832519, 832583)
(838249, 838349)
(844777, 844841)
(883807, 883871)
(912103, 912167)
(919447, 919511)
(954763, 954827)
(981823, 981887)
(997813, 997877)
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: formatting io kernel lists lists.lazy math math.functions
math.primes.lists sequences ;

: adj-primes ( -- list ) lprimes dup cdr lzip ;

: diff ( pair -- n ) first2 swap - ;

: adj-primes-diff ( -- list )
adj-primes [ dup diff suffix ] lmap-lazy ;

: big-adj-primes-diff ( -- list )
adj-primes-diff [ last 36 > ] lfilter ;

: square? ( n -- ? ) sqrt dup >integer number= ;

: big-sq-adj-primes-diff ( -- list )
big-adj-primes-diff [ last square? ] lfilter ;

"Adjacent primes under a million whose difference is a square > 36:" print nl
"p1       p2       difference" print
"============================" print
big-sq-adj-primes-diff [ second 1,000,000 < ] lwhile
[ "%-6d   %-6d   %d\n" vprintf ] leach
```
Output:
```Adjacent primes under a million whose difference is a square > 36:

p1       p2       difference
============================
89689    89753    64
107377   107441   64
288583   288647   64
367957   368021   64
381103   381167   64
396733   396833   100
400759   400823   64
445363   445427   64
623107   623171   64
625699   625763   64
637003   637067   64
710713   710777   64
725209   725273   64
779413   779477   64
801883   801947   64
803749   803813   64
821677   821741   64
832519   832583   64
838249   838349   100
844777   844841   64
883807   883871   64
912103   912167   64
919447   919511   64
954763   954827   64
981823   981887   64
997813   997877   64
```

## Fermat

```Func Issqr( n ) = if (Sqrt(n))^2=n then 1 else 0 fi.;
i:=3;
j:=3;
while j<1000000 do
j:=i+2;
while j < 1000000 do
if Isprime(j) then
if j-i>36 and Issqr(j-i) then !!(i,j,j-i) fi;
i:=j;
fi;
j:=j+2;
od;
od;```

## FreeBASIC

```#include "isprime.bas"

function nextprime( n as uinteger ) as uinteger
'finds the next prime after n
if n = 0 then return 2
if n < 3 then return n + 1
dim as integer q = n + 2
while not isprime(q)
q+=2
wend
return q
end function

function issquare( n as uinteger ) as boolean
if int(sqr(n))^2 = n then return true else return false
end function

dim as uinteger i=3, j=0
while j<1000000
j = nextprime(i)
if j-i > 36 and issquare(j-i) then print i, j, j-i
i = j
wend
```
Output:
```
89689         89753         64
107377        107441        64
288583        288647        64
367957        368021        64
381103        381167        64
396733        396833        100
400759        400823        64
445363        445427        64
623107        623171        64
625699        625763        64
637003        637067        64
710713        710777        64
725209        725273        64
779413        779477        64
801883        801947        64
803749        803813        64
821677        821741        64
832519        832583        64
838249        838349        100
844777        844841        64
883807        883871        64
912103        912167        64
919447        919511        64
954763        954827        64
981823        981887        64
997813        997877        64

```

## Go

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

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

func main() {
limit := 999999
primes := rcu.Primes(limit)
fmt.Println("Adjacent primes under 1,000,000 whose difference is a square > 36:")
for i := 1; i < len(primes); i++ {
diff := primes[i] - primes[i-1]
if diff > 36 {
s := int(math.Sqrt(float64(diff)))
if diff == s*s {
cp1 := rcu.Commatize(primes[i])
cp2 := rcu.Commatize(primes[i-1])
fmt.Printf("%7s - %7s = %3d = %2d x %2d\n", cp1, cp2, diff, s, s)
}
}
}
}
```
Output:
```Same as Wren example.
```

## GW-BASIC

```10 P=3 : P2=0
20 GOSUB 180
30 IF P2>1000000! THEN END
40 R = P2-P
50 IF R > 36 AND INT(SQR(R))^2=R THEN PRINT P,P2,R
60 P=P2
70 GOTO 20
80 REM tests if a number is prime
90 Q=0
100 IF P = 2 THEN Q = 1:RETURN
110 IF P=3 THEN Q=1:RETURN
120 I=1
130 I=I+1
140 IF INT(P/I)*I = P THEN RETURN
150 IF I*I<=P THEN GOTO 130
160 Q = 1
170 RETURN
180 REM finds the next prime after P, result in P2
190 IF P = 0 THEN P2 = 2: RETURN
200 IF P<3 THEN P2 = P + 1: RETURN
210 T = P
220 P = P + 1
230 GOSUB 80
240 IF Q = 1 THEN P2 = P: P = T: RETURN
250 GOTO 220
```

## J

```   #(,.-~/"1) p:0 1+/~I.(= <.)6.5>.%:2-~/\p:i.p:inv 1e6  NB. count them
26
(,.-~/"1) p:0 1+/~I.(= <.)6.5>.%:2-~/\p:i.p:inv 1e6   NB. show them
89689  89753  64
107377 107441  64
288583 288647  64
367957 368021  64
381103 381167  64
396733 396833 100
400759 400823  64
445363 445427  64
623107 623171  64
625699 625763  64
637003 637067  64
710713 710777  64
725209 725273  64
779413 779477  64
801883 801947  64
803749 803813  64
821677 821741  64
832519 832583  64
838249 838349 100
844777 844841  64
883807 883871  64
912103 912167  64
919447 919511  64
954763 954827  64
981823 981887  64
997813 997877  64
```

In other words: enumerate primes less than 1e6, find the pairwise differences, find where the prime pairs where maximum of their square root and 6.5 is an integer, and list those pairs with their differences.

## 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.

Preliminaries

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

# Primes less than . // infinite
def primes:
(. // infinite) as \$n
| if \$n < 3 then empty
else 2, (range(3; \$n) | select(is_prime))
end;```

```# Input is given to primes/0  - to determine the maximum prime to consider
# Output: stream of [\$prime, \$nextPrime]
def isSquare: sqrt | . == floor;

foreach primes as \$p ( {previous: null};
.emit = null
| if .previous != null
and ((\$p - .previous) | isSquare)
then .emit = [.previous, \$p]
else .
end
| .previous = \$p;
select(.emit).emit);

# Input is given to primes/0 to determine the maximum prime to consider.
# Gap must be greater than \$gap
"Adjacent primes under \(.) whose difference is a square > \(\$gap):",
| (.[1] - .[0]) as \$diff
| select(\$diff > \$gap)
| "\(.[1]|l) - \(.[0]|l) = \(\$diff|lpad(4))" ) ;

Output:

As for #ALGOL_68.

## Julia

```using Primes

function squareprimegaps(limit)
pri = primes(limit)
squares = Set([1; [x * x for x in 2:2:100]])
diffs = [pri[i] - pri[i - 1] for i in 2:length(pri)]
squarediffs = sort(unique(filter(n -> n in squares, diffs)))
println("\n\nSquare prime gaps to \$limit:")
for sq in squarediffs
i = findfirst(x -> x == sq, diffs)
n = count(x -> x == sq, diffs)
if limit == 1000000 && sq > 36
println("Showing all \$n with square difference \$sq:")
pairs = [(pri[i], pri[i + 1]) for i in findall(x -> x == sq, diffs)]
foreach(p -> print(last(p), first(p) % 4 == 0 ? "\n" : " "), enumerate(pairs))
else
println("Square difference \$sq: \$n found. Example: (\$(pri[i]), \$(pri[i + 1])).")
end
end
end

squareprimegaps(1_000_000)
squareprimegaps(10_000_000_000)
```
Output:
```
Square prime gaps to 1000000:
Square difference 1: 1 found. Example: (2, 3).
Square difference 4: 8143 found. Example: (7, 11).
Square difference 16: 2881 found. Example: (1831, 1847).
Square difference 36: 767 found. Example: (9551, 9587).
Showing all 24 with square difference 64:
(89689, 89753) (107377, 107441) (288583, 288647) (367957, 368021)
(381103, 381167) (400759, 400823) (445363, 445427) (623107, 623171)
(625699, 625763) (637003, 637067) (710713, 710777) (725209, 725273)
(779413, 779477) (801883, 801947) (803749, 803813) (821677, 821741)
(832519, 832583) (844777, 844841) (883807, 883871) (912103, 912167)
(919447, 919511) (954763, 954827) (981823, 981887) (997813, 997877)
Showing all 2 with square difference 100:
(396733, 396833) (838249, 838349)

Square prime gaps to 10000000000:
Square difference 1: 1 found. Example: (2, 3).
Square difference 4: 27409998 found. Example: (7, 11).
Square difference 16: 15888305 found. Example: (1831, 1847).
Square difference 36: 11593345 found. Example: (9551, 9587).
Square difference 64: 1434957 found. Example: (89689, 89753).
Square difference 100: 268933 found. Example: (396733, 396833).
Square difference 144: 35563 found. Example: (11981443, 11981587).
Square difference 196: 1254 found. Example: (70396393, 70396589).
Square difference 256: 41 found. Example: (1872851947, 1872852203).
```

## Mathematica/Wolfram Language

```ps = Prime[Range[PrimePi[10^6]]];
ps = Partition[ps, 2, 1];
ps = {#1, #2, #2 - #1} & @@@ ps;
ps //= Select[Extract[{3}]/*GreaterThan[36]];
ps //= Select[Extract[{3}]/*Sqrt/*IntegerQ];
ps // Grid
```
Output:
```89689	89753	64
107377	107441	64
288583	288647	64
367957	368021	64
381103	381167	64
396733	396833	100
400759	400823	64
445363	445427	64
623107	623171	64
625699	625763	64
637003	637067	64
710713	710777	64
725209	725273	64
779413	779477	64
801883	801947	64
803749	803813	64
821677	821741	64
832519	832583	64
838249	838349	100
844777	844841	64
883807	883871	64
912103	912167	64
919447	919511	64
954763	954827	64
981823	981887	64
997813	997877	64```

## PARI/GP

`for(i=3,1000000,j=nextprime(i+1);if(isprime(i)&&j-i>36&&issquare(j-i),print(i," ",j," ",j-i)))`

## Perl

```#!/usr/bin/perl

use warnings;
use ntheory qw( primes is_square );

my \$primeref = primes(1e6);
for my \$i (1 .. \$#\$primeref) {
(my \$diff = \$primeref->[\$i] - \$primeref->[\$i - 1]) > 36 or next;
is_square(\$diff) and print "\$primeref->[\$i] - \$primeref->[\$i - 1] = \$diff\n";
}
```
Output:
```89753 - 89689 = 64
107441 - 107377 = 64
288647 - 288583 = 64
368021 - 367957 = 64
381167 - 381103 = 64
396833 - 396733 = 100
400823 - 400759 = 64
445427 - 445363 = 64
623171 - 623107 = 64
625763 - 625699 = 64
637067 - 637003 = 64
710777 - 710713 = 64
725273 - 725209 = 64
779477 - 779413 = 64
801947 - 801883 = 64
803813 - 803749 = 64
821741 - 821677 = 64
832583 - 832519 = 64
838349 - 838249 = 100
844841 - 844777 = 64
883871 - 883807 = 64
912167 - 912103 = 64
919511 - 919447 = 64
954827 - 954763 = 64
981887 - 981823 = 64
997877 - 997813 = 64
```

## Phix

```with javascript_semantics
constant limit = 1_000_000
sequence primes = get_primes_le(limit),
square = repeat(false,floor(sqrt(limit)))
integer sq = 7
while sq*sq<=length(square) do
square[sq*sq] = true
sq += 1
end while
for i=2 to length(primes) do
integer p = primes[i],
q = primes[i-1],
d = p-q
if square[d] then
printf(1,"%6d - %6d = %d\n",{p,q,d})
end if
end for
```
Output:
``` 89753 -  89689 = 64
107441 - 107377 = 64
288647 - 288583 = 64
368021 - 367957 = 64
381167 - 381103 = 64
396833 - 396733 = 100
400823 - 400759 = 64
445427 - 445363 = 64
623171 - 623107 = 64
625763 - 625699 = 64
637067 - 637003 = 64
710777 - 710713 = 64
725273 - 725209 = 64
779477 - 779413 = 64
801947 - 801883 = 64
803813 - 803749 = 64
821741 - 821677 = 64
832583 - 832519 = 64
838349 - 838249 = 100
844841 - 844777 = 64
883871 - 883807 = 64
912167 - 912103 = 64
919511 - 919447 = 64
954827 - 954763 = 64
981887 - 981823 = 64
997877 - 997813 = 64
```

## Python

```import math
print("working...")
limit = 1000000
Primes = []
oldPrime = 0
newPrime = 0
x = 0

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(x):
if (x == n*n):
return 1
return 0

for n in range(limit):
if isPrime(n):
Primes.append(n)

for n in range(2,len(Primes)):
pr1 = Primes[n]
pr2 = Primes[n-1]
diff = pr1 - pr2
flag = issquare(diff)
if (flag == 1 and diff > 36):
print(str(pr1) + " " + str(pr2) + " diff = " + str(diff))

print("done...")
```
Output:
```working...
89753 89689 diff = 64
107441 107377 diff = 64
288647 288583 diff = 64
368021 367957 diff = 64
381167 381103 diff = 64
396833 396733 diff = 100
400823 400759 diff = 64
445427 445363 diff = 64
623171 623107 diff = 64
625763 625699 diff = 64
637067 637003 diff = 64
710777 710713 diff = 64
725273 725209 diff = 64
779477 779413 diff = 64
801947 801883 diff = 64
803813 803749 diff = 64
821741 821677 diff = 64
832583 832519 diff = 64
838349 838249 diff = 100
844841 844777 diff = 64
883871 883807 diff = 64
912167 912103 diff = 64
919511 919447 diff = 64
954827 954763 diff = 64
981887 981823 diff = 64
997877 997813 diff = 64
done...
```

## Raku

```use Lingua::EN::Numbers;
use Math::Primesieve;

my \$iterator = Math::Primesieve::iterator.new;
my \$limit    = 1e10;
my @squares  = (1..30).map: *²;
my \$last     = 2;
my @gaps;
my @counts;

loop {
my \$this = (my \$p = \$iterator.next) - \$last;
quietly @gaps[\$this].push(\$last) if +@gaps[\$this] < 10;
@counts[\$this]++;
last if \$p > \$limit;
\$last = \$p;
}

print "Adjacent primes up to {comma \$limit.Int} with a gap value that is a perfect square:";
for @gaps.pairs.grep: { (.key ∈ @squares) && .value.defined} -> \$p {
my \$ten = (@counts[\$p.key] > 10) ?? ', (first ten)' !! '';
say "\nGap {\$p.key}: {comma @counts[\$p.key]} found\$ten:";
put join "\n", \$p.value.batch(5)».map({"(\$_, {\$_+ \$p.key})"})».join(', ');
}
```
Output:
```Adjacent primes up to 10,000,000,000 with a gap value that is a perfect square:
Gap 1: 1 found:
(2, 3)

Gap 4: 27,409,998 found, (first ten):
(7, 11), (13, 17), (19, 23), (37, 41), (43, 47)
(67, 71), (79, 83), (97, 101), (103, 107), (109, 113)

Gap 16: 15,888,305 found, (first ten):
(1831, 1847), (1933, 1949), (2113, 2129), (2221, 2237), (2251, 2267)
(2593, 2609), (2803, 2819), (3121, 3137), (3373, 3389), (3391, 3407)

Gap 36: 11,593,345 found, (first ten):
(9551, 9587), (12853, 12889), (14107, 14143), (15823, 15859), (18803, 18839)
(22193, 22229), (22307, 22343), (22817, 22853), (24281, 24317), (27143, 27179)

Gap 64: 1,434,957 found, (first ten):
(89689, 89753), (107377, 107441), (288583, 288647), (367957, 368021), (381103, 381167)
(400759, 400823), (445363, 445427), (623107, 623171), (625699, 625763), (637003, 637067)

Gap 100: 268,933 found, (first ten):
(396733, 396833), (838249, 838349), (1313467, 1313567), (1648081, 1648181), (1655707, 1655807)
(2345989, 2346089), (2784373, 2784473), (3254959, 3255059), (3595489, 3595589), (4047157, 4047257)

Gap 144: 35,563 found, (first ten):
(11981443, 11981587), (18687587, 18687731), (20024339, 20024483), (20388583, 20388727), (21782503, 21782647)
(25507423, 25507567), (27010003, 27010147), (28716287, 28716431), (31515413, 31515557), (32817493, 32817637)

Gap 196: 1,254 found, (first ten):
(70396393, 70396589), (191186251, 191186447), (208744777, 208744973), (233987851, 233988047), (288568771, 288568967)
(319183093, 319183289), (336075937, 336076133), (339408151, 339408347), (345247753, 345247949), (362956201, 362956397)

Gap 256: 41 found, (first ten):
(1872851947, 1872852203), (2362150363, 2362150619), (2394261637, 2394261893), (2880755131, 2880755387), (2891509333, 2891509589)
(3353981623, 3353981879), (3512569873, 3512570129), (3727051753, 3727052009), (3847458487, 3847458743), (4008610423, 4008610679)```

## Ring

```load "stdlib.ring"
see "working..." + nl
limit = 1000000
Primes = []
oldPrime = 0
newPrime = 0
x = 0

for n = 1 to limit
if isprime(n)
ok
next

for n = 2 to len(Primes)
pr1 = Primes[n]
pr2 = Primes[n-1]
diff = pr1 - pr2
flag = issquare(diff)
if flag = 1 and diff > 36
see "" + pr1 + " " + pr2 + " diff = " + diff + nl
ok
next

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...
89753 89689 diff = 64
107441 107377 diff = 64
288647 288583 diff = 64
368021 367957 diff = 64
381167 381103 diff = 64
396833 396733 diff = 100
400823 400759 diff = 64
445427 445363 diff = 64
623171 623107 diff = 64
625763 625699 diff = 64
637067 637003 diff = 64
710777 710713 diff = 64
725273 725209 diff = 64
779477 779413 diff = 64
801947 801883 diff = 64
803813 803749 diff = 64
821741 821677 diff = 64
832583 832519 diff = 64
838349 838249 diff = 100
844841 844777 diff = 64
883871 883807 diff = 64
912167 912103 diff = 64
919511 919447 diff = 64
954827 954763 diff = 64
981887 981823 diff = 64
997877 997813 diff = 64
done...
```

## Ruby

```require "prime"

Prime.each(1_000_000).each_cons(2) do |a, b|
diff = b - a
next unless diff > 36
isqrt = Integer.sqrt(diff)
puts "#{b} - #{a} = #{diff}" if isqrt*isqrt == diff
end
```
Output:
```89753 - 89689 = 64
107441 - 107377 = 64
288647 - 288583 = 64
368021 - 367957 = 64
381167 - 381103 = 64
396833 - 396733 = 100
400823 - 400759 = 64
445427 - 445363 = 64
623171 - 623107 = 64
625763 - 625699 = 64
637067 - 637003 = 64
710777 - 710713 = 64
725273 - 725209 = 64
779477 - 779413 = 64
801947 - 801883 = 64
803813 - 803749 = 64
821741 - 821677 = 64
832583 - 832519 = 64
838349 - 838249 = 100
844841 - 844777 = 64
883871 - 883807 = 64
912167 - 912103 = 64
919511 - 919447 = 64
954827 - 954763 = 64
981887 - 981823 = 64
997877 - 997813 = 64
```

## Sidef

```var p = 2
var upto = 1e6

each_prime(p.next_prime, upto, {|q|
if (q-p > 36 && is_square(q-p)) {
say "#{'%6s' % q} - #{'%6s' % p} = #{'%2s' % isqrt(q-p)}^2"
}
p = q
})
```
Output:
``` 89753 -  89689 =  8^2
107441 - 107377 =  8^2
288647 - 288583 =  8^2
368021 - 367957 =  8^2
381167 - 381103 =  8^2
396833 - 396733 = 10^2
400823 - 400759 =  8^2
445427 - 445363 =  8^2
623171 - 623107 =  8^2
625763 - 625699 =  8^2
637067 - 637003 =  8^2
710777 - 710713 =  8^2
725273 - 725209 =  8^2
779477 - 779413 =  8^2
801947 - 801883 =  8^2
803813 - 803749 =  8^2
821741 - 821677 =  8^2
832583 - 832519 =  8^2
838349 - 838249 = 10^2
844841 - 844777 =  8^2
883871 - 883807 =  8^2
912167 - 912103 =  8^2
919511 - 919447 =  8^2
954827 - 954763 =  8^2
981887 - 981823 =  8^2
997877 - 997813 =  8^2
```

## Wren

Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

var limit = 1e6 - 1
var primes = Int.primeSieve(limit)
System.print("Adjacent primes under 1,000,000 whose difference is a square > 36:")
for (i in 1...primes.count) {
var diff = primes[i] - primes[i-1]
if (diff > 36) {
var s = diff.sqrt.floor
if (diff == s * s) {
Fmt.print ("\$,7d - \$,7d = \$3d = \$2d x \$2d", primes[i], primes[i-1], diff, s, s)
}
}
}
```
Output:
```Adjacent primes under 1,000,000 whose difference is a square > 36:
89,753 -  89,689 =  64 =  8 x  8
107,441 - 107,377 =  64 =  8 x  8
288,647 - 288,583 =  64 =  8 x  8
368,021 - 367,957 =  64 =  8 x  8
381,167 - 381,103 =  64 =  8 x  8
396,833 - 396,733 = 100 = 10 x 10
400,823 - 400,759 =  64 =  8 x  8
445,427 - 445,363 =  64 =  8 x  8
623,171 - 623,107 =  64 =  8 x  8
625,763 - 625,699 =  64 =  8 x  8
637,067 - 637,003 =  64 =  8 x  8
710,777 - 710,713 =  64 =  8 x  8
725,273 - 725,209 =  64 =  8 x  8
779,477 - 779,413 =  64 =  8 x  8
801,947 - 801,883 =  64 =  8 x  8
803,813 - 803,749 =  64 =  8 x  8
821,741 - 821,677 =  64 =  8 x  8
832,583 - 832,519 =  64 =  8 x  8
838,349 - 838,249 = 100 = 10 x 10
844,841 - 844,777 =  64 =  8 x  8
883,871 - 883,807 =  64 =  8 x  8
912,167 - 912,103 =  64 =  8 x  8
919,511 - 919,447 =  64 =  8 x  8
954,827 - 954,763 =  64 =  8 x  8
981,887 - 981,823 =  64 =  8 x  8
997,877 - 997,813 =  64 =  8 x  8
```

## XPL0

```func IsPrime(N);        \Return 'true' if odd N > 2 is prime
int  N, I;
[for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false;
I:= I+1;
];
return true;
];

int N, P0, P1, D, RD;
[P0:= 2;
for N:= 3 to 1_000_000-1 do
[if IsPrime(N) then
[P1:= N;
D:= P1 - P0;            \D is even because odd - odd = even
if D >= 64 then         \the next even square > 36 is 64
[RD:= sqrt(D);
if RD*RD = D then
[IntOut(0, P1);  Text(0, " - ");
IntOut(0, P0);  Text(0, " = ");
IntOut(0, D);  CrLf(0);
];
];
P0:= P1;
];
N:= N+1;                    \step by 1+1 = 2 (for odd numbers)
];
]```
Output:
```89753 - 89689 = 64
107441 - 107377 = 64
288647 - 288583 = 64
368021 - 367957 = 64
381167 - 381103 = 64
396833 - 396733 = 100
400823 - 400759 = 64
445427 - 445363 = 64
623171 - 623107 = 64
625763 - 625699 = 64
637067 - 637003 = 64
710777 - 710713 = 64
725273 - 725209 = 64
779477 - 779413 = 64
801947 - 801883 = 64
803813 - 803749 = 64
821741 - 821677 = 64
832583 - 832519 = 64
838349 - 838249 = 100
844841 - 844777 = 64
883871 - 883807 = 64
912167 - 912103 = 64
919511 - 919447 = 64
954827 - 954763 = 64
981887 - 981823 = 64
997877 - 997813 = 64
```