# Wilson primes of order n

Wilson primes of order n
You are encouraged to solve this task according to the task description, using any language you may know.
Definition

A Wilson prime of order n is a prime number   p   such that   p2   exactly divides:

```     (n − 1)! × (p − n)! − (− 1)n
```

If   n   is   1,   the latter formula reduces to the more familiar:   (p - n)! + 1   where the only known examples for   p   are   5,   13,   and   563.

Calculate and show on this page the Wilson primes, if any, for orders n = 1 to 11 inclusive and for primes p < 18   or,
if your language supports big integers, for p < 11,000.

## ALGOL 68

Translation of: Visual Basic .NET

... but using a sieve for primeallity checking.
As with the various BASIC samples, all calculations are done MOD p2 so arbitrary precision integers are not needed.

```BEGIN # find Wilson primes of order n, primes such that:                    #
#                    ( ( n - 1 )! x ( p - n )! - (-1)^n ) mod p^2 = 0 #
PR read "primes.incl.a68" PR                  # include prime utilities #
[]BOOL primes = PRIMESIEVE 11 000;         # sieve the primes to 11 500 #
# returns TRUE if p is an nth order Wilson prime                        #
PROC is wilson = ( INT n, p )BOOL:
IF p < n THEN FALSE
ELSE
LONG INT p2    = p * p;
LONG INT prod := 1;
FOR i TO n - 1 DO # prod := ( n - 1 )! MOD p2                    #
prod := ( prod * i ) MOD p2
OD;
FOR i TO p - n DO # prod := ( ( p - n )! * ( n - 1 )! ) MOD p2   #
prod := ( prod * i ) MOD p2
OD;
0 = ( p2 + prod + IF ODD n THEN 1 ELSE -1 FI ) MOD p2
FI # is wilson # ;
# find the Wilson primes #
print( ( " n: Wilson primes", newline ) );
print( ( "-----------------", newline ) );
FOR n TO 11 DO
print( ( whole( n, -2 ), ":" ) );
IF is wilson( n, 2 ) THEN print( ( " 2" ) ) FI;
FOR p FROM 3 BY 2 TO UPB primes DO
IF primes[ p ] THEN
IF is wilson( n, p ) THEN print( ( " ", whole( p, 0 ) ) ) FI
FI
OD;
print( ( newline ) )
OD
END```
Output:
``` n: Wilson primes
-----------------
1: 5 13 563
2: 2 3 11 107 4931
3: 7
4: 10429
5: 5 7 47
6: 11
7: 17
8:
9: 541
10: 11 1109
11: 17 2713
```

## BASIC

### Applesoft BASIC

Translation of: Chipmunk Basic
```100 home
110 print "n:     Wilson primes"
120 print "--------------------"
130 for n = 1 to 11
140  print n;chr\$(9);
150  for p = 2 to 18
160   gosub 240
170   if pt = 0 then goto 200
180   gosub 340
190   if wnpt = 1 then print p,
200  next p
210  print
220 next n
230 end
240 rem tests if the number P is prime
250 rem result is stored in PT
260 pt = 1
270 if p = 2 then return
280 if p * 2 - int(p / 2) = 0 then pt = 0 : return
290 j = 3
300 if j*j > p then return
310 if p * j - int(p / j) = 0 then pt = 0 : return
320 j = j+2
330 goto 300
340 rem tests if the prime p is a Wilson prime of order n
350 rem make sure it actually is prime first
360 rem result is stored in wnpt
370 wnpt = 0
380 if p = 2 and n = 2 then wnpt = 1 : return
390 if n > p then wnpt = 0 : return
400 prod = 1 : p2 = p*p
410 for i = 1 to n-1
420  prod = (prod*i) : gosub 500
430 next i
440 for i = 1 to p-n
450  prod = (prod*i) : gosub 500
460 next i
470 prod = (p2+prod-(-1)^n) : gosub 500
480 if prod = 0 then wnpt = 1 : return
490 wnpt = 0 : return
500 rem prod mod p2 fails if prod > 32767 so brew our own modulus function
510 prod = prod-int(prod/p2)*p2
520 return
```

### BASIC256

Translation of: FreeBASIC
```function isPrime(v)
if v <= 1 then return False
for i = 2 To int(sqr(v))
if v % i = 0 then return False
next i
return True
end function

function isWilson(n, p)
if p < n then return false
prod = 1
p2 = p*p #p^2
for i = 1 to n-1
prod = (prod*i) mod p2
next i
for i = 1 to p-n
prod = (prod*i) mod p2
next i
prod = (p2 + prod - (-1)**n) mod p2
if prod = 0 then return true else return false
end function

print " n:      Wilson primes"
print "----------------------"
for n = 1 to 11
print n;" :  ";
for p = 3 to 10499 step 2
if isPrime(p) and isWilson(n, p) then print p; " ";
next p
print
next n
end```

### Chipmunk Basic

Works with: Chipmunk Basic version 3.6.4
Works with: GW-BASIC
Works with: MSX_BASIC
Works with: PC-BASIC version any
Works with: QBasic
Translation of: GW-BASIC
```100 cls
110 print "n:     Wilson primes"
120 print "--------------------"
130 for n = 1 to 11
140  print n;chr\$(9);
150  for p = 2 to 18
160   gosub 240
170   if pt = 0 then goto 200
180   gosub 340
190   if wnpt = 1 then print p,
200  next p
210  print
220 next n
230 end
240 rem tests if the number P is prime
250 rem result is stored in PT
260 pt = 1
270 if p = 2 then return
280 if p mod 2 = 0 then pt = 0 : return
290 j = 3
300 if j*j > p then return
310 if p mod j = 0 then pt = 0 : return
320 j = j+2
330 goto 300
340 rem tests if the prime p is a Wilson prime of order n
350 rem make sure it actually is prime first
360 rem result is stored in wnpt
370 wnpt = 0
380 if p = 2 and n = 2 then wnpt = 1 : return
390 if n > p then wnpt = 0 : return
400 prod = 1 : p2 = p*p
410 for i = 1 to n-1
420  prod = (prod*i) : gosub 500
430 next i
440 for i = 1 to p-n
450  prod = (prod*i) : gosub 500
460 next i
470 prod = (p2+prod-(-1)^n) : gosub 500
480 if prod = 0 then wnpt = 1 : return
490 wnpt = 0 : return
500 rem prod mod p2 fails if prod > 32767 so brew our own modulus function
510 prod = prod-int(prod/p2)*p2
520 return
```

### QBasic

Works with: QBasic
Works with: QuickBasic
Translation of: FreeBASIC
```FUNCTION isPrime (ValorEval)
IF ValorEval < 2 THEN isPrime = False
IF ValorEval MOD 2 = 0 THEN isPrime = 2
IF ValorEval MOD 3 = 0 THEN isPrime = 3
d = 5
WHILE d * d <= ValorEval
IF ValorEval MOD d = 0 THEN isPrime = False ELSE d = d + 2
WEND
isPrime = True
END FUNCTION

FUNCTION isWilson (n, p)
IF p < n THEN isWilson = False
prod = 1
p2 = p ^ 2
FOR i = 1 TO n - 1
prod = (prod * i) MOD p2
NEXT i
FOR i = 1 TO p - n
prod = (prod * i) MOD p2
NEXT i
prod = (p2 + prod - (-1) ^ n) MOD p2
IF prod = 0 THEN isWilson = True ELSE isWilson = False
END FUNCTION

PRINT " n:      Wilson primes"
PRINT "----------------------"
FOR n = 1 TO 11
PRINT USING "##:      "; n;
FOR p = 3 TO 10099 STEP 2
If isPrime(p) AND isWilson(n, p) Then Print p; " ";
NEXT p
PRINT
NEXT n
END
```

### MSX Basic

Both the GW-BASIC and Chipmunk Basic solutions work without change.

### Visual Basic .NET

Translation of: QBasic

...but includes 2 and the 4th order Wilson Prime.

```Option Strict On
Option Explicit On

Module WilsonPrimes

Function isPrime(p As Integer) As Boolean
If p < 2 Then Return False
If p Mod 2 = 0 Then Return p = 2
IF p Mod 3 = 0 Then Return p = 3
Dim d As Integer = 5
Do While d * d <= p
If p Mod d = 0 Then
Return False
Else
d = d + 2
End If
Loop
Return True
End Function

Function isWilson(n As Integer, p As Integer) As Boolean
If p < n Then Return False
Dim prod As Long = 1
Dim p2 As Long = p * p
For i = 1 To n - 1
prod = (prod * i) Mod p2
Next i
For i = 1 To p - n
prod = (prod * i) Mod p2
Next i
prod = (p2 + prod - If(n Mod 2 = 0, 1, -1)) Mod p2
Return prod = 0
End Function

Sub Main()
Console.Out.WriteLine(" n:      Wilson primes")
Console.Out.WriteLine("----------------------")
For n = 1 To 11
Console.Out.Write(n.ToString.PadLeft(2) & ":      ")
If isWilson(n, 2) Then Console.Out.Write("2 ")
For p = 3 TO 10499 Step 2
If isPrime(p) And isWilson(n, p) Then Console.Out.Write(p & " ")
Next p
Console.Out.WriteLine()
Next n
End Sub

End Module
```
Output:
``` n:      Wilson primes
----------------------
1:      5 13 563
2:      2 3 11 107 4931
3:      7
4:      10429
5:      5 7 47
6:      11
7:      17
8:
9:      541
10:      11 1109
11:      17 2713
```

### Yabasic

Translation of: FreeBASIC
```print "n:      Wilson primes"
print "---------------------"
for n = 1 to 11
print n, ":",
for p = 3 to 10099 step 2
if isPrime(p) and isWilson(n, p) then print p, " ", : fi
next p
print
next n
end

sub isPrime(v)
if v < 2 then return False : fi
if mod(v, 2) = 0 then return v = 2 : fi
if mod(v, 3) = 0 then return v = 3 : fi
d = 5
while d * d <= v
if mod(v, d) = 0 then return False else d = d + 2 : fi
end while
return True
end sub

sub isWilson(n, p)
if p < n then return False : fi
prod = 1
p2 = p**2
for i = 1 to n-1
prod = mod((prod*i), p2)
next i
for i = 1 to p-n
prod = mod((prod*i), p2)
next i
prod = mod((p2 + prod - (-1)**n), p2)
if prod = 0 then return True else return False : fi
end sub```

## C

Translation of: Wren
Library: GMP
```#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <gmp.h>

bool *sieve(int limit) {
int i, p;
limit++;
// True denotes composite, false denotes prime.
bool *c = calloc(limit, sizeof(bool)); // all false by default
c[0] = true;
c[1] = true;
for (i = 4; i < limit; i += 2) c[i] = true;
p = 3; // Start from 3.
while (true) {
int p2 = p * p;
if (p2 >= limit) break;
for (i = p2; i < limit; i += 2 * p) c[i] = true;
while (true) {
p += 2;
if (!c[p]) break;
}
}
return c;
}

int main() {
const int limit = 11000;
int i, j, n, pc = 0;
unsigned long p;
bool *c = sieve(limit);
for (i = 0; i < limit; ++i) {
if (!c[i]) ++pc;
}
unsigned long *primes = (unsigned long *)malloc(pc * sizeof(unsigned long));
for (i = 0, j = 0; i < limit; ++i) {
if (!c[i]) primes[j++] = i;
}
mpz_t *facts = (mpz_t *)malloc(limit *sizeof(mpz_t));
for (i = 0; i < limit; ++i) mpz_init(facts[i]);
mpz_set_ui(facts[0], 1);
for (i = 1; i < limit; ++i) mpz_mul_ui(facts[i], facts[i-1], i);
mpz_t f, sign;
mpz_init(f);
mpz_init_set_ui(sign, 1);
printf(" n:  Wilson primes\n");
printf("--------------------\n");
for (n = 1; n < 12; ++n) {
printf("%2d:  ", n);
mpz_neg(sign, sign);
for (i = 0; i < pc; ++i) {
p = primes[i];
if (p < n) continue;
mpz_mul(f, facts[n-1], facts[p-n]);
mpz_sub(f, f, sign);
if (mpz_divisible_ui_p(f, p*p)) printf("%ld ", p);
}
printf("\n");
}
free(c);
free(primes);
for (i = 0; i < limit; ++i) mpz_clear(facts[i]);
free(facts);
return 0;
}
```
Output:
``` n:  Wilson primes
--------------------
1:  5 13 563
2:  2 3 11 107 4931
3:  7
4:  10429
5:  5 7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713
```

## C++

Library: GMP
```#include <iomanip>
#include <iostream>
#include <vector>
#include <gmpxx.h>

std::vector<int> generate_primes(int limit) {
std::vector<bool> sieve(limit >> 1, true);
for (int p = 3, s = 9; s < limit; p += 2) {
if (sieve[p >> 1]) {
for (int q = s; q < limit; q += p << 1)
sieve[q >> 1] = false;
}
s += (p + 1) << 2;
}
std::vector<int> primes;
if (limit > 2)
primes.push_back(2);
for (int i = 1; i < sieve.size(); ++i) {
if (sieve[i])
primes.push_back((i << 1) + 1);
}
return primes;
}

int main() {
using big_int = mpz_class;
const int limit = 11000;
std::vector<big_int> f{1};
f.reserve(limit);
big_int factorial = 1;
for (int i = 1; i < limit; ++i) {
factorial *= i;
f.push_back(factorial);
}
std::vector<int> primes = generate_primes(limit);
std::cout << " n | Wilson primes\n--------------------\n";
for (int n = 1, s = -1; n <= 11; ++n, s = -s) {
std::cout << std::setw(2) << n << " |";
for (int p : primes) {
if (p >= n && (f[n - 1] * f[p - n] - s) % (p * p) == 0)
std::cout << ' ' << p;
}
std::cout << '\n';
}
}
```
Output:
``` n | Wilson primes
--------------------
1 | 5 13 563
2 | 2 3 11 107 4931
3 | 7
4 | 10429
5 | 5 7 47
6 | 11
7 | 17
8 |
9 | 541
10 | 11 1109
11 | 17 2713
```

## EasyLang

Translation of: FreeBASIC
```func isprim num .
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
func is_wilson n p .
if p < n
return 0
.
prod = 1
p2 = p * p
for i = 1 to n - 1
prod = prod * i mod p2
.
for i = 1 to p - n
prod = prod * i mod p2
.
prod = (p2 + prod - pow -1 n) mod p2
if prod = 0
return 1
.
return 0
.
print "n:  Wilson primes"
print "-----------------"
for n = 1 to 11
write n & "   "
for p = 3 step 2 to 10099
if isprim p = 1 and is_wilson n p = 1
write p & " "
.
.
print ""
.
```

## F#

This task uses Extensible Prime Generator (F#)

```// Wilson primes. Nigel Galloway: July 31st., 2021
let rec fN g=function n when n<2I->g |n->fN(n*g)(n-1I)
let fG (n:int)(p:int)=let g,p=bigint n,bigint p in (((fN 1I (g-1I))*(fN 1I (p-g))-(-1I)**n)%(p*p))=0I
[1..11]|>List.iter(fun n->printf "%2d -> " n; let fG=fG n in pCache|>Seq.skipWhile((>)n)|>Seq.takeWhile((>)11000)|>Seq.filter fG|>Seq.iter(printf "%d "); printfn "")
```
Output:
``` 1 -> 5 13 563
2 -> 2 3 11 107 4931
3 -> 7
4 -> 10429
5 -> 5 7 47
6 -> 11
7 -> 17
8 ->
9 -> 541
10 -> 11 1109
11 -> 17 2713
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: formatting infix io kernel literals math math.functions
math.primes math.ranges prettyprint sequences sequences.extras ;

<< CONSTANT: limit 11,000 >>

CONSTANT: primes \$[ limit primes-upto ]

CONSTANT: factorials
\$[ limit [1,b] 1 [ * ] accumulate* 1 prefix ]

: factorial ( n -- n! ) factorials nth ; inline

INFIX:: fn ( p n -- m )
factorial(n-1) * factorial(p-n) - -1**n ;

: wilson? ( p n -- ? ) [ fn ] keepd sq divisor? ; inline

: order ( n -- seq )
primes swap [ [ < ] curry drop-while ] keep
[ wilson? ] curry filter ;

: order. ( n -- )
dup "%2d:  " printf order [ pprint bl ] each nl ;

" n:  Wilson primes\n--------------------" print
11 [1,b] [ order. ] each
```
Output:
``` n:  Wilson primes
--------------------
1:  5 13 563
2:  2 3 11 107 4931
3:  7
4:  10429
5:  5 7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713
```

## FreeBASIC

This excludes the trivial case p=n=2.

```#include "isprime.bas"

function is_wilson( n as uinteger, p as uinteger ) as boolean
'tests if p^2 divides (n-1)!(p-n)! - (-1)^n
'does NOT test the primality of p; do that first before you call this!
'using mods no big nums are required
if p<n then return false
dim as uinteger prod = 1, i, p2 = p^2
for i = 1 to n-1
prod = (prod*i) mod p2
next i
for i = 1 to p-n
prod = (prod*i) mod p2
next i
prod = (p2 + prod - (-1)^n) mod p2
if prod = 0 then return true else return false
end function

print "n:      Wilson primes"
print "--------------------"
for n as uinteger = 1 to 11
print using "##      ";n;
for p as uinteger = 3 to 10099 step 2
if isprime(p) andalso is_wilson(n, p) then print p;" ";
next p
print
next n
```
Output:
```
n:      Wilson primes
--------------------
1      5 13 563
2      3 11 107 4931
3      7
4
5      5 7 47
6      11
7      17
8
9      541
10      11 1109
11      17 2713

```

## Go

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

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

func main() {
const LIMIT = 11000
primes := rcu.Primes(LIMIT)
facts := make([]*big.Int, LIMIT)
facts[0] = big.NewInt(1)
for i := int64(1); i < LIMIT; i++ {
facts[i] = new(big.Int)
facts[i].Mul(facts[i-1], big.NewInt(i))
}
sign := int64(1)
f := new(big.Int)
zero := new(big.Int)
fmt.Println(" n:  Wilson primes")
fmt.Println("--------------------")
for n := 1; n < 12; n++ {
fmt.Printf("%2d:  ", n)
sign = -sign
for _, p := range primes {
if p < n {
continue
}
f.Mul(facts[n-1], facts[p-n])
f.Sub(f, big.NewInt(sign))
p2 := int64(p * p)
bp2 := big.NewInt(p2)
if f.Rem(f, bp2).Cmp(zero) == 0 {
fmt.Printf("%d ", p)
}
}
fmt.Println()
}
}
```
Output:
``` n:  Wilson primes
--------------------
1:  5 13 563
2:  2 3 11 107 4931
3:  7
4:  10429
5:  5 7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713
```

## GW-BASIC

```10 PRINT "n:     Wilson primes"
20 PRINT "--------------------"
30 FOR N = 1 TO 11
40 PRINT USING "##";N;
50 FOR P=2 TO 18
60 GOSUB 140
70 IF PT=0 THEN GOTO 100
80 GOSUB 230
90 IF WNPT=1 THEN PRINT P;
100 NEXT P
110 PRINT
120 NEXT N
130 END
140 REM tests if the number P is prime
150 REM result is stored in PT
160 PT = 1
170 IF P=2 THEN RETURN
175 IF P MOD 2 = 0 THEN PT=0:RETURN
180 J=3
190 IF J*J>P THEN RETURN
200 IF P MOD J = 0 THEN PT = 0: RETURN
210 J = J + 2
220 GOTO 190
230 REM tests if the prime P is a Wilson prime of order N
240 REM make sure it actually is prime first
250 REM RESULT is stored in WNPT
260 WNPT=0
270 IF P=2 AND N=2 THEN WNPT = 1: RETURN
280 IF N>P THEN WNPT=0: RETURN
290 PROD# = 1 : P2 = P*P
300 FOR I = 1 TO N-1
310 PROD# = (PROD#*I) : GOSUB 3000
320 NEXT I
330 FOR I = 1 TO P-N
340 PROD# = (PROD#*I) : GOSUB 3000
350 NEXT I
360 PROD# = (P2+PROD#-(-1)^N) : GOSUB 3000
370 IF PROD# = 0 THEN WNPT = 1: RETURN
380 WNPT = 0: RETURN
3000 REM    PROD# MOD P2 fails if PROD#>32767 so brew our own modulus function
3010 PROD# = PROD# - INT(PROD#/P2)*P2
3020 RETURN
```

## J

```   wilson=. 0 = (*:@] | _1&^@[ -~ -~ *&! <:@[)^:<:

(>: i. 11x) ([ ;"0 wilson"0/ <@# ]) i.&.(p:inv) 11000
┌──┬───────────────┐
│1 │5 13 563       │
├──┼───────────────┤
│2 │2 3 11 107 4931│
├──┼───────────────┤
│3 │7              │
├──┼───────────────┤
│4 │10429          │
├──┼───────────────┤
│5 │5 7 47         │
├──┼───────────────┤
│6 │11             │
├──┼───────────────┤
│7 │17             │
├──┼───────────────┤
│8 │               │
├──┼───────────────┤
│9 │541            │
├──┼───────────────┤
│10│11 1109        │
├──┼───────────────┤
│11│17 2713        │
└──┴───────────────┘
```

## Java

```import java.math.BigInteger;
import java.util.*;

public class WilsonPrimes {
public static void main(String[] args) {
final int limit = 11000;
BigInteger[] f = new BigInteger[limit];
f[0] = BigInteger.ONE;
BigInteger factorial = BigInteger.ONE;
for (int i = 1; i < limit; ++i) {
factorial = factorial.multiply(BigInteger.valueOf(i));
f[i] = factorial;
}
List<Integer> primes = generatePrimes(limit);
System.out.printf(" n | Wilson primes\n--------------------\n");
BigInteger s = BigInteger.valueOf(-1);
for (int n = 1; n <= 11; ++n) {
System.out.printf("%2d |", n);
for (int p : primes) {
if (p >= n && f[n - 1].multiply(f[p - n]).subtract(s)
.mod(BigInteger.valueOf(p * p))
.equals(BigInteger.ZERO))
System.out.printf(" %d", p);
}
s = s.negate();
System.out.println();
}
}

private static List<Integer> generatePrimes(int limit) {
boolean[] sieve = new boolean[limit >> 1];
Arrays.fill(sieve, true);
for (int p = 3, s = 9; s < limit; p += 2) {
if (sieve[p >> 1]) {
for (int q = s; q < limit; q += p << 1)
sieve[q >> 1] = false;
}
s += (p + 1) << 2;
}
List<Integer> primes = new ArrayList<>();
if (limit > 2)
for (int i = 1; i < sieve.length; ++i) {
if (sieve[i])
primes.add((i << 1) + 1);
}
return primes;
}
}
```
Output:
``` n | Wilson primes
--------------------
1 | 5 13 563
2 | 2 3 11 107 4931
3 | 7
4 | 10429
5 | 5 7 47
6 | 11
7 | 17
8 |
9 | 541
10 | 11 1109
11 | 17 2713
```

## jq

Works with jq (*)

Works with gojq, the Go implementation of jq

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

(*) The C implementation of jq lacks the precision for handling the case p<11,000 so the output below is based on a run of gojq.

Preliminaries

```def emit_until(cond; stream): label \$out | stream | if cond then break \$out else . end;

# For 0 <= \$n <= ., factorials[\$n] is \$n !
def factorials:
reduce range(1; .+1) as \$n ([1];
.[\$n] = \$n * .[\$n-1]);

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

def primes: 2, (range(3; infinite; 2) | select(is_prime));```

Wilson primes

```# Input: the limit of \$p
def wilson_primes:
def sgn: if . % 2 == 0 then 1 else -1 end;

. as \$limit
| factorials as \$facts
| " n:  Wilson primes\n--------------------",
(range(1;12) as \$n
| "\(\$n|lpad(2)) :  \(
[emit_until( . >= \$limit; primes)
| select(. as \$p
| \$p >= \$n and
((\$facts[\$n - 1] * \$facts[\$p - \$n] - (\$n|sgn))
% (\$p*\$p) == 0 )) ])" );

11000 | wilson_primes```
Output:

gojq -ncr -f rc-wilson-primes.jq

``` n:  Wilson primes
--------------------
1 :  [5,13,563]
2 :  [2,3,11,107,4931]
3 :  [7]
4 :  [10429]
5 :  [5,7,47]
6 :  [11]
7 :  [17]
8 :  []
9 :  [541]
10 :  [11,1109]
11 :  [17,2713]
```

## Julia

Translation of: Wren
```using Primes

function wilsonprimes(limit = 11000)
sgn, facts = 1, accumulate(*, 1:limit, init = big"1")
println(" n:  Wilson primes\n--------------------")
for n in 1:11
print(lpad(n, 2), ":  ")
sgn = -sgn
for p in primes(limit)
if p > n && (facts[n < 2 ? 1 : n - 1] * facts[p - n] - sgn) % p^2 == 0
print("\$p ")
end
end
println()
end
end

wilsonprimes()
```

Output: Same as Wren example.

## Mathematica /Wolfram Language

```ClearAll[WilsonPrime]
WilsonPrime[n_Integer] := Module[{primes, out},
primes = Prime[Range[PrimePi[11000]]];
out = Reap@Do[
If[Divisible[((n - 1)!) ((p - n)!) - (-1)^n, p^2], Sow[p]]
,
{p, primes}
];
First[out[[2]], {}]
]
Do[
Print[WilsonPrime[n]]
,
{n, 1, 11}
]
```
Output:
```{5,13,563}
{2,3,11,107,4931}
{7}
{10429}
{5,7,47}
{11}
{17}
{}
{541}
{11,1109}
{17,2713}```

## Nim

Translation of: Go
Library: bignum

As in Nim there is not (not yet?) a standard module to deal with big numbers, we use the third party module “bignum”.

```import strformat, strutils
import bignum

const Limit = 11_000

# Build list of primes using "nextPrime" function from "bignum".
var primes: seq[int]
var p = newInt(2)
while p < Limit:
p = p.nextPrime()

# Build list of factorials.
var facts: array[Limit, Int]
facts[0] = newInt(1)
for i in 1..<Limit:
facts[i] = facts[i - 1] * i

var sign = 1
echo " n: Wilson primes"
echo "—————————————————"
for n in 1..11:
sign = -sign
var wilson: seq[int]
for p in primes:
if p < n: continue
let f = facts[n - 1] * facts[p - n] - sign
if f mod (p * p) == 0:
echo &"{n:2}:  ", wilson.join(" ")
```
Output:
```n: Wilson primes
—————————————————
1:  5 13 563
2:  2 3 11 107 4931
3:  7
4:  10429
5:  5 7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713```

## PARI/GP

Translation of: Julia
```default("parisizemax", "1024M");

\\ Define the function wilsonprimes with a default limit of 11000
wilsonprimes(limit) = {
\\ Set the default limit if not specified
my(limit = if(limit, limit, 11000));
\\ Precompute factorial values up to the limit to save time
my(facts = vector(limit, i, i!));
\\ Sign variable for adjustment in the formula
my(sgn = 1);
print(" n:  Wilson primes\n--------------------");
\\ Loop over the specified range (1 to 11 in the original code)
for(n = 1, 11,
print1(Str(" ", n, ":  "));
sgn = -sgn; \\ Toggle the sign
\\ Loop over all primes up to the limit
forprime(p = 2, limit,
\\ Check the Wilson prime condition modified for PARI/GP
index=1;
if(n<2,index=1,index=n-1);
if(p > n && Mod(facts[index] * facts[p - n] - sgn, p^2) == 0,
print1(Str(p, " "));
)
);
print1("\n");
);
}

\\ Execute the function with the default limit
wilsonprimes();```
Output:
``` n:  Wilson primes
--------------------
1:  5 13 563
2:  3 11 107 4931
3:  7
4:  10429
5:  7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713

```

## Perl

Library: ntheory
```use strict;
use warnings;
use ntheory <primes factorial>;

my @primes = @{primes( 10500 )};

for my \$n (1..11) {
printf "%3d: %s\n", \$n, join ' ', grep { \$_ >= \$n && 0 == (factorial(\$n-1) * factorial(\$_-\$n) - (-1)**\$n) % \$_**2 } @primes
}
```
Output:
```  1: 5 13 563
2: 2 3 11 107 4931
3: 7
4: 10429
5: 5 7 47
6: 11
7: 17
8:
9: 541
10: 11 1109
11: 17 2713```

## Phix

Translation of: Wren
```with javascript_semantics
constant limit = 11000
include mpfr.e
mpz f = mpz_init()
sequence primes = get_primes_le(limit),
facts = mpz_inits(limit,1) -- (nb 0!==1!, same slot)
for i=2 to limit do mpz_mul_si(facts[i],facts[i-1],i) end for
integer sgn = 1
printf(1," n:  Wilson primes\n")
printf(1,"--------------------\n")
for n=1 to 11 do
printf(1,"%2d:  ", n)
sgn = -sgn
for i=1 to length(primes) do
integer p = primes[i]
if p>=n then
mpz_mul(f,facts[max(n-1,1)],facts[max(p-n,1)])
mpz_sub_si(f,f,sgn)
if mpz_divisible_ui_p(f,p*p) then
printf(1,"%d ", p)
end if
end if
end for
printf(1,"\n")
end for
```

Output: Same as Wren example.

## Prolog

Works with: SWI Prolog
```main:-
wilson_primes(11000).

wilson_primes(Limit):-
writeln('  n | Wilson primes\n---------------------'),
make_factorials(Limit),
find_prime_numbers(Limit),
wilson_primes(1, 12, -1).

wilson_primes(N, N, _):-!.
wilson_primes(N, M, S):-
wilson_primes(N, S),
S1 is -S,
N1 is N + 1,
wilson_primes(N1, M, S1).

wilson_primes(N, S):-
writef('%3r |', [N]),
N1 is N - 1,
factorial(N1, F1),
is_prime(P),
P >= N,
PN is P - N,
factorial(PN, F2),
0 is (F1 * F2 - S) mod (P * P),
writef(' %w', [P]),
fail.
wilson_primes(_, _):-
nl.

make_factorials(N):-
retractall(factorial(_, _)),
make_factorials(N, 0, 1).

make_factorials(N, N, F):-
assert(factorial(N, F)),
!.
make_factorials(N, M, F):-
assert(factorial(M, F)),
M1 is M + 1,
F1 is F * M1,
make_factorials(N, M1, F1).
```

Module for finding prime numbers up to some limit:

```:- module(prime_numbers, [find_prime_numbers/1, is_prime/1]).
:- dynamic is_prime/1.

find_prime_numbers(N):-
retractall(is_prime(_)),
assertz(is_prime(2)),
init_sieve(N, 3),
sieve(N, 3).

init_sieve(N, P):-
P > N,
!.
init_sieve(N, P):-
assertz(is_prime(P)),
Q is P + 2,
init_sieve(N, Q).

sieve(N, P):-
P * P > N,
!.
sieve(N, P):-
is_prime(P),
!,
S is P * P,
cross_out(S, N, P),
Q is P + 2,
sieve(N, Q).
sieve(N, P):-
Q is P + 2,
sieve(N, Q).

cross_out(S, N, _):-
S > N,
!.
cross_out(S, N, P):-
retract(is_prime(S)),
!,
Q is S + 2 * P,
cross_out(Q, N, P).
cross_out(S, N, P):-
Q is S + 2 * P,
cross_out(Q, N, P).
```
Output:
```  n | Wilson primes
---------------------
1 | 5 13 563
2 | 2 3 11 107 4931
3 | 7
4 | 10429
5 | 5 7 47
6 | 11
7 | 17
8 |
9 | 541
10 | 11 1109
11 | 17 2713
```

## Python

```# wilson_prime.py by xing216
def sieve(n):
multiples = []
for i in range(2, n+1):
if i not in multiples:
yield i
for j in range(i*i, n+1, i):
multiples.append(j)
def intListToString(list):
return " ".join([str(i) for i in list])
limit = 11000
primes = list(sieve(limit))
facs = [1]
for i in range(1,limit):
facs.append(facs[-1]*i)
sign = 1
print(" n: Wilson primes")
print("—————————————————")

for n in range(1,12):
sign = -sign
wilson = []
for p in primes:
if p < n: continue
f = facs[n-1] * facs[p-n] - sign
if f % p**2 == 0: wilson.append(p)
print(f"{n:2d}: {intListToString(wilson)}")
```
Output:
``` n: Wilson primes
—————————————————
1: 5 13 563
2: 2 3 11 107 4931
3: 7
4: 10429
5: 5 7 47
6: 11
7: 17
8:
9: 541
10: 11 1109
11: 17 2713
```

## Racket

```#lang racket

(require math/number-theory)

(define ((wilson-prime? n) p)
(and (>= p n)
(prime? p)
(divides? (sqr p)
(- (* (factorial (- n 1))
(factorial (- p n)))
(expt -1 n)))))

(define primes<11000 (filter prime? (range 1 11000)))

(for ((n (in-range 1 (add1 11))))
(printf "~a: ~a~%" n (filter (wilson-prime? n) primes<11000)))
```
Output:
```1: (5 13 563)
2: (2 3 11 107 4931)
3: (7)
4: (10429)
5: (5 7 47)
6: (11)
7: (17)
8: ()
9: (541)
10: (11 1109)
11: (17 2713)```

## Raku

```# Factorial
sub postfix:<!> (Int \$n) { (constant f = 1, |[\×] 1..*)[\$n] }

# Invisible times
sub infix:<⁢> is tighter(&infix:<**>) { \$^a * \$^b };

# Prime the iterator for thread safety
sink 11000!;

my @primes = ^1.1e4 .grep: *.is-prime;

say
'  n: Wilson primes
────────────────────';

.say for (1..40).hyper(:1batch).map: -> \𝒏 {
sprintf "%3d: %s", 𝒏, @primes.grep( -> \𝒑 { (𝒑 ≥ 𝒏) && ((𝒏 - 1)!⁢(𝒑 - 𝒏)! - (-1) ** 𝒏) %% 𝒑² } ).Str
}
```
Output:
```  n: Wilson primes
────────────────────
1: 5 13 563
2: 2 3 11 107 4931
3: 7
4: 10429
5: 5 7 47
6: 11
7: 17
8:
9: 541
10: 11 1109
11: 17 2713
12:
13: 13
14:
15: 349
16: 31
17: 61 251 479
18:
19: 71
20: 59 499
21:
22:
23:
24: 47 3163
25:
26:
27: 53
28: 347
29:
30: 137 1109 5179
31:
32: 71
33: 823 1181 2927
34: 149
35: 71
36:
37: 71 1889
38:
39: 491
40: 59 71 1171```

## REXX

For more (extended) results,   see this task's discussion page.

```/*REXX program finds and displays Wilson primes:  a prime   P   such that  P**2 divides:*/
/*────────────────── (n-1)! * (P-n)!  -  (-1)**n   where  n  is 1 ──◄ 11,   and  P < 18.*/
parse arg oLO oHI hip .                          /*obtain optional argument from the CL.*/
if  oLO=='' |  oLO==","  then  oLO=      1       /*Not specified?  Then use the default.*/
if  oHI=='' |  oHI==","  then  oHI=     11       /* "      "         "   "   "     "    */
if  hip=='' |  hip==","  then  hip=  11000       /* "      "         "   "   "     "    */
call genP                                        /*build array of semaphores for primes.*/
!!.= .                                           /*define the  default  for factorials. */
bignum= !(hip)                                   /*calculate a ginormous factorial prod.*/
parse value bignum 'E0' with ex 'E' ex .         /*obtain possible exponent of factorial*/
numeric digits (max(9, ex+2) )                   /*calculate max # of dec. digits needed*/
call facts hip                                   /*go & calculate a number of factorials*/
title= ' Wilson primes  P  of order '  oLO " ──► " oHI',  where  P < '  commas(hip)
w= length(title) + 1                             /*width of columns of possible numbers.*/
say ' order │'center(title, w     )
say '───────┼'center(""   , w, '─')
do n=oLO  to oHI;  nf= !(n-1)              /*precalculate a factorial product.    */
z= -1**n                                   /*     "       " plus or minus (+1│-1).*/
if n==1   then lim= 103                    /*limit to known primes for 1st order. */
else lim=   #                    /*  "    "  all     "    "  orders ≥ 2.*/
\$=                                         /*\$:  a line (output) of Wilson primes.*/
do j=1  for lim;    p= @.j              /*search through some generated primes.*/
if (nf*!(p-n)-z)//sq.j\==0 then iterate /*expression ~ q.j ?  No, then skip it.*/       /* ◄■■■■■■■ the filter.*/
\$= \$  ' '  commas(p)                    /*add a commatized prime  ──►  \$  list.*/
end   /*p*/

if \$==''  then \$= '        (none found within the range specified)'
say center(n, 7)'│'  substr(\$, 2)          /*display what Wilson primes we found. */
end   /*n*/
say '───────┴'center(""   , w, '─')
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!:      arg x; if !!.x\==.  then return !!.x;  a=1;  do f=1  for x;   a=a*f; end; return a
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
facts:  !!.= 1;   x= 1;  do  f=1  for hip;   x= x * f;   !!.f= x;           end;  return
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP:        @.1=2; @.2=3; @.3=5; @.4=7;  @.5=11 /*define some low primes.              */
!.=0;  !.2=1; !.3=1; !.5=1; !.7=1;  !.11=1 /*   "     "   "    "     semaphores.  */
sq.1=4; sq.2=9; sq.3= 25; sq.4= 49; #= 5;  sq.#= @.#**2   /*squares of low primes.*/
do j=@.#+2  by 2  for max(0, hip%2-@.#%2-1)     /*find odd primes from here on. */
parse var  j   ''  -1  _;  if    _==5  then iterate    /*J ÷ 5?   (right digit).*/
if j//3==0  then iterate;  if j//7==0  then iterate    /*" " 3?    Is J ÷ by 7? */
do k=5  while sq.k<=j             /* [↓]  divide by the known odd primes.*/
if j // @.k == 0  then iterate j  /*Is  J ÷ X?  Then not prime.     ___  */
end   /*k*/                       /* [↑]  only process numbers  ≤  √ J   */
#= #+1;    @.#= j;    sq.#= j*j;  !.j= 1 /*bump # of Ps; assign next P;  P²; P# */
end          /*j*/;               return
```
output   when using the default inputs:
``` order │ Wilson primes  P  of order  1  ──►  11,  where  P <  11,000
───────┼─────────────────────────────────────────────────────────────
1   │   5   13   563
2   │   2   3   11   107   4,931
3   │   7
4   │   10,429
5   │   5   7   47
6   │   11
7   │   17
8   │        (none found within the range specified)
9   │   541
10   │   11   1,109
11   │   17   2,713
───────┴─────────────────────────────────────────────────────────────
```

## RPL

Works with: RPL version HP-49C
```« → maxp
« { }
1 11 FOR n
{ } n
IF DUP ISPRIME? NOT THEN NEXTPRIME END
WHILE DUP maxp < REPEAT
n 1 - FACT OVER n - FACT * -1 n ^ -
IF OVER SQ MOD NOT THEN SWAP OVER + SWAP END
NEXTPRIME
END
DROP 1 →LIST +
NEXT
» » 'TASK' STO
```
Output:
```1: { { 5 13 } { 2 3 11 } { 7 } { } { 5 7 } { 11 } { 17 } { } { } { 11 } { 17 } }
```

## Ruby

```require "prime"

module Modulo
refine Integer do
def factorial_mod(m) = (1..self).inject(1){|prod, n| (prod *= n) % m }
end
end

using Modulo
primes = Prime.each(11000).to_a

(1..11).each do |n|
res = primes.select do |pr|
prpr = pr*pr
((n-1).factorial_mod(prpr) * (pr-n).factorial_mod(prpr) - (-1)**n) % (prpr) == 0
end
puts "#{n.to_s.rjust(2)}: #{res.inspect}"
end
```
Output:
``` 1: [5, 13, 563]
2: [2, 3, 11, 107, 4931]
3: [7]
4: [10429]
5: [5, 7, 47]
6: [11]
7: [17]
8: []
9: [541]
10: [11, 1109]
11: [17, 2713]

```

## Rust

```// [dependencies]
// rug = "1.13.0"

use rug::Integer;

fn generate_primes(limit: usize) -> Vec<usize> {
let mut sieve = vec![true; limit >> 1];
let mut p = 3;
let mut sq = p * p;
while sq < limit {
if sieve[p >> 1] {
let mut q = sq;
while q < limit {
sieve[q >> 1] = false;
q += p << 1;
}
}
sq += (p + 1) << 2;
p += 2;
}
let mut primes = Vec::new();
if limit > 2 {
primes.push(2);
}
for i in 1..sieve.len() {
if sieve[i] {
primes.push((i << 1) + 1);
}
}
primes
}

fn factorials(limit: usize) -> Vec<Integer> {
let mut f = vec![Integer::from(1)];
let mut factorial = Integer::from(1);
f.reserve(limit);
for i in 1..limit {
factorial *= i as u64;
f.push(factorial.clone());
}
f
}

fn main() {
let limit = 11000;
let f = factorials(limit);
let primes = generate_primes(limit);
println!(" n | Wilson primes\n--------------------");
let mut s = -1;
for n in 1..=11 {
print!("{:2} |", n);
for p in &primes {
if *p >= n {
let mut num = Integer::from(&f[n - 1] * &f[*p - n]);
num -= s;
if num % ((p * p) as u64) == 0 {
print!(" {}", p);
}
}
}
println!();
s = -s;
}
}
```
Output:
``` n | Wilson primes
--------------------
1 | 5 13 563
2 | 2 3 11 107 4931
3 | 7
4 | 10429
5 | 5 7 47
6 | 11
7 | 17
8 |
9 | 541
10 | 11 1109
11 | 17 2713
```

## Sidef

```func is_wilson_prime(p, n = 1) {
var m = p*p
(factorialmod(n-1, m) * factorialmod(p-n, m) - (-1)**n) % m == 0
}

var primes = 1.1e4.primes

say "  n: Wilson primes\n────────────────────"

for n in (1..11) {
printf("%3d: %s\n", n, primes.grep {|p| is_wilson_prime(p, n) })
}
```
Output:
```  n: Wilson primes
────────────────────
1: [5, 13, 563]
2: [2, 3, 11, 107, 4931]
3: [7]
4: [10429]
5: [5, 7, 47]
6: [11]
7: [17]
8: []
9: [541]
10: [11, 1109]
11: [17, 2713]
```

## Wren

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

var limit = 11000
var primes = Int.primeSieve(limit)
var facts = List.filled(limit, null)
facts[0] = BigInt.one
for (i in 1...11000) facts[i] = facts[i-1] * i
var sign = 1
System.print(" n:  Wilson primes")
System.print("--------------------")
for (n in 1..11) {
Fmt.write("\$2d:  ", n)
sign = -sign
for (p in primes) {
if (p < n) continue
var f = facts[n-1] * facts[p-n] - sign
if (f.isDivisibleBy(p*p)) Fmt.write("%(p) ", p)
}
System.print()
}
```
Output:
``` n:  Wilson primes
--------------------
1:  5 13 563
2:  2 3 11 107 4931
3:  7
4:  10429
5:  5 7 47
6:  11
7:  17
8:
9:  541
10:  11 1109
11:  17 2713
```