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# Primality by Wilson's theorem

Primality by Wilson's theorem
You are encouraged to solve this task according to the task description, using any language you may know.

Write a boolean function that tells whether a given integer is prime using Wilson's theorem.

By Wilson's theorem, a number p is prime if and only if p divides `(p - 1)! + 1`.

Remember that 1 and all non-positive integers are not prime.

## 11l

Translation of: Python
`F is_wprime(Int64 n)   R n > 1 & (n == 2 | (n % 2 & (factorial(n - 1) + 1) % n == 0)) V c = 20print(‘Primes under #.:’.format(c), end' "\n  ")print((0 .< c).filter(n -> is_wprime(n)))`
Output:
```Primes under 20:
[2, 3, 5, 7, 11, 13, 17, 19]
```

## 8086 Assembly

`	cpu	8086	org	100hsection	.text	jmp	demo	;;;	Wilson primality test of CX.	;;;	Zero flag set if CX prime. Destroys AX, BX, DX.wilson:	xor	ax,ax		; AX will hold intermediate fac-mod value	inc	ax	mov	bx,cx		; BX = factorial loop counter	dec	bx.loop:	mul	bx		; DX:AX = AX*BX	div	cx		; modulus goes in DX	mov	ax,dx	dec	bx		; Next value	jnz	.loop		; If not zero yet, go again	inc	ax		; fac-mod + 1 equal to input?	cmp	ax,cx		; Set flags according to result	ret	;;;	Demo: print primes under 256demo:	mov	cx,2.loop:	call	wilson		; Is it prime?	jnz	.next		; If not, try next number	mov	ax,cx	call	print		; Otherwise, print the number.next:	inc	cl		; Next number.	jnz	.loop		; If <256, try next number	ret	;;;	Print value in AX using DOS syscallprint:	mov	bp,10		; Divisor	mov	bx,numbuf	; Pointer to buffer.digit:	xor	dx,dx	div	bp		; Divide AX and get digit in DX	add	dl,'0'		; Make ASCII	dec	bx		; Store in buffer	mov	[bx],dl	test	ax,ax		; Done yet?	jnz	.digit		; If not, get next digit	mov	dx,bx		; Print buffer	mov	ah,9		; 9 = MS-DOS syscall to print a string	int	21h	retsection	.data	db	'*****'		; Space to hold ASCII number for outputnumbuf:	db	13,10,'\$'`
Output:
```2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
127
131
137
139
149
151
157
163
167
173
179
181
191
193
197
199
211
223
227
229
233
239
241
251
```

`---- Determine primality using Wilon's theorem.-- Uses the approach from Algol W -- allowing large primes without the use of big numbers.--with Ada.Text_IO; use Ada.Text_IO; procedure Main is   type u_64 is mod 2**64;   package u_64_io is new modular_io (u_64);   use u_64_io;    function Is_Prime (n : u_64) return Boolean is      fact_Mod_n : u_64 := 1;   begin      if n < 2 then         return False;      end if;      for i in 2 .. n - 1 loop         fact_Mod_n := (fact_Mod_n * i) rem n;      end loop;      return fact_Mod_n = n - 1;   end Is_Prime;    num : u_64 := 1;   type cols is mod 12;   count : cols := 0;begin   while num < 500 loop      if Is_Prime (num) then         if count = 0 then            New_Line;         end if;         Put (Item => num, Width => 6);         count := count + 1;      end if;      num := num + 1;   end loop;end Main;  `
Output:
```     2     3     5     7    11    13    17    19    23    29    31    37
41    43    47    53    59    61    67    71    73    79    83    89
97   101   103   107   109   113   127   131   137   139   149   151
157   163   167   173   179   181   191   193   197   199   211   223
227   229   233   239   241   251   257   263   269   271   277   281
283   293   307   311   313   317   331   337   347   349   353   359
367   373   379   383   389   397   401   409   419   421   431   433
439   443   449   457   461   463   467   479   487   491   499
```

## ALGOL 68

Translation of: ALGOL W

As with many samples on this page, applies the modulo operation at each step in calculating the factorial, to avoid needing large integeres.

`BEGIN    # find primes using Wilson's theorem:                               #    #    p is prime if ( ( p - 1 )! + 1 ) mod p = 0                     #     # returns true if p is a prime by Wilson's theorem, false otherwise #    #         computes the factorial mod p at each stage, so as to      #    #         allow numbers whose factorial won't fit in 32 bits        #    PROC is wilson prime = ( INT p )BOOL:         BEGIN            INT factorial mod p := 1;            FOR i FROM 2 TO p - 1 DO factorial mod p *:= i MODAB p OD;            factorial mod p = p - 1         END # is wilson prime # ;     FOR i TO 100 DO IF is wilson prime( i ) THEN print( ( " ", whole( i, 0 ) ) ) FI ODEND`
Output:
``` 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## ALGOL W

As with the APL, Tiny BASIC and other samples, this computes the factorials mod p at each multiplication to avoid needing numbers larger than the 32 bit limit.

`begin    % find primes using Wilson's theorem:                               %    %    p is prime if ( ( p - 1 )! + 1 ) mod p = 0                     %     % returns true if n is a prime by Wilson's theorem, false otherwise %    %         computes the factorial mod p at each stage, so as to      %    %         allow numbers whose factorial won't fit in 32 bits        %    logical procedure isWilsonPrime ( integer value n ) ;    begin        integer factorialModN;        factorialModN := 1;        for i := 2 until n - 1 do factorialModN := ( factorialModN * i ) rem n;        factorialModN = n - 1    end isWilsonPrime ;     for i := 1 until 100 do if isWilsonPrime( i ) then writeon( i_w := 1, s_w := 0, " ", i );end.`
Output:
```2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## APL

This version avoids huge intermediate values by calculating the modulus after each step of the factorial multiplication. This is necessary for the function to work correctly with more than the first few numbers.

`wilson ← {⍵<2:0 ⋄ (⍵-1)=(⍵|×)/⍳⍵-1}`
Output:
```      wilson {(⍺⍺¨⍵)/⍵} ⍳200
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163
167 173 179 181 191 193 197 199```

The naive version (using APL's built-in factorial) looks like this:

`naiveWilson ← {⍵<2:0 ⋄ 0=⍵|1+!⍵-1}`

But due to loss of precision with large floating-point values, it only works correctly up to number 19 even with ⎕CT set to zero:

Output:
```      ⎕CT←0 ⋄ naiveWilson {(⍺⍺¨⍵)/⍵} ⍳20
2 3 5 7 11 13 17 19 20```

## AppleScript

Nominally, the AppleScript solution would be as follows, the 'mod n' at every stage of the factorial being to keep the numbers within the range the language can handle:

`on isPrime(n)    if (n < 2) then return false    set f to n - 1    repeat with i from (n - 2) to 2 by -1        set f to f * i mod n    end repeat     return ((f + 1) mod n = 0)end isPrime local output, nset output to {}repeat with n from 0 to 500    if (isPrime(n)) then set end of output to nend repeatoutput`
Output:
`{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499}`

In fact, though, the modding by n after every multiplication means there are only three possibilities for the final value of f: n - 1 (if n's a prime), 2 (if n's 4), or 0 (if n's any other non-prime). So the test at the end of the handler could be simplified. Another thing is that if f becomes 0 at some point in the repeat, it obviously stays that way for the remaining iterations, so quite a bit of time can be saved by testing for it and returning false immediately if it occurs. And if 2 and its multiples are caught before the repeat, any other non-prime will guarantee a jump out of the handler. Simply reaching the end will mean n's a prime.

It turns out too that false results only occur when multiplying numbers between √n and n - √n and that only multiplying numbers in this range still leads to the correct outcomes. And if this isn't abusing Wilson's theorem enough, multiples of 2 and 3 can be prechecked and omitted from the "factorial" process altogether, much as they can be skipped in tests for primality by trial division:

`on isPrime(n)    -- Check for numbers < 2 and 2 & 3 and their multiples.    if (n < 4) then return (n > 1)    if ((n mod 2 = 0) or (n mod 3 = 0)) then return false    -- Only multiply numbers in the range √n -> n - √n that are 1 less and 1 more than multiples of 6,    -- starting with a number that's 1 less than a multiple of 6 and as close as practical to √n.    tell (n ^ 0.5 div 1) to set f to it - (it - 2) mod 6 + 3    repeat with i from f to (n - f - 6) by 6        set f to f * i mod n * (i + 2) mod n        if (f = 0) then return false    end repeat     return trueend isPrime`

## Arturo

`factorial: function [x]-> product 1..x wprime?: function [n][    if n < 2 -> return false    zero? mod add factorial sub n 1 1 n] print "Primes below 20 via Wilson's theorem:"print select 1..20 => wprime?`
Output:
```Primes below 20 via Wilson's theorem:
2 3 5 7 11 13 17 19```

## AWK

` # syntax: GAWK -f PRIMALITY_BY_WILSONS_THEOREM.AWK# converted from FreeBASICBEGIN {    start = 2    stop = 200    for (i=start; i<=stop; i++) {      if (is_wilson_prime(i)) {        printf("%5d%1s",i,++count%10?"":"\n")      }    }    printf("\nWilson primality test range %d-%d: %d\n",start,stop,count)    exit(0)}function is_wilson_prime(n,  fct,i) {    fct = 1    for (i=2; i<=n-1; i++) {      # because (a mod n)*b = (ab mod n)      # it is not necessary to calculate the entire factorial      fct = (fct * i) % n    }    return(fct == n-1)} `
Output:
```    2     3     5     7    11    13    17    19    23    29
31    37    41    43    47    53    59    61    67    71
73    79    83    89    97   101   103   107   109   113
127   131   137   139   149   151   157   163   167   173
179   181   191   193   197   199
Wilson primality test range 2-200: 46
```

## C

`#include <stdbool.h>#include <stdint.h>#include <stdio.h> uint64_t factorial(uint64_t n) {    uint64_t product = 1;     if (n < 2) {        return 1;    }     for (; n > 0; n--) {        uint64_t prev = product;        product *= n;        if (product < prev) {            fprintf(stderr, "Overflowed\n");            return product;        }    }     return product;} // uses wilson's theorembool isPrime(uint64_t n) {    uint64_t large = factorial(n - 1) + 1;    return (large % n) == 0;} int main() {    uint64_t n;     // Can check up to 21, more will require a big integer library    for (n = 2; n < 22; n++) {        printf("Is %llu prime: %d\n", n, isPrime(n));    }     return 0;}`
Output:
```Is 2 prime: 1
Is 3 prime: 1
Is 4 prime: 0
Is 5 prime: 1
Is 6 prime: 0
Is 7 prime: 1
Is 8 prime: 0
Is 9 prime: 0
Is 10 prime: 0
Is 11 prime: 1
Is 12 prime: 0
Is 13 prime: 1
Is 14 prime: 0
Is 15 prime: 0
Is 16 prime: 0
Is 17 prime: 1
Is 18 prime: 0
Is 19 prime: 1
Is 20 prime: 0
Is 21 prime: 0```

## C#

Performance comparison to Sieve of Eratosthenes.

`using System;using System.Linq;using System.Collections;using static System.Console;using System.Collections.Generic;using BI = System.Numerics.BigInteger; class Program {   // initialization    const int fst = 120, skp = 1000, max = 1015; static double et1, et2; static DateTime st;    static string ms1 = "Wilson's theorem method", ms2 = "Sieve of Eratosthenes method",        fmt = "--- {0} ---\n\nThe first {1} primes are:", fm2 = "{0} prime thru the {1} prime:";    static List<int> lst = new List<int>();   // dumps a chunk of the prime list (lst)    static void Dump(int s, int t, string f) {        foreach (var item in lst.Skip(s).Take(t)) Write(f, item); WriteLine("\n"); }   // returns the ordinal string representation of a number    static string Ord(int x, string fmt = "{0:n0}") {      var y = x % 10; if ((x % 100) / 10 == 10 || y > 3) y = 0;      return string.Format(fmt, x) + "thstndrd".Substring(y << 1, 2); }   // shows the results of one type of prime tabulation    static void ShowOne(string title, ref double et) {        WriteLine(fmt, title, fst); Dump(0, fst, "{0,-3} ");        WriteLine(fm2, Ord(skp), Ord(max)); Dump(skp - 1, max - skp + 1, "{0,4} ");        WriteLine("Time taken: {0}ms\n", et = (DateTime.Now - st).TotalMilliseconds); }   // for stand-alone computation    static BI factorial(int n) { BI res = 1; if (n < 2) return res;        while (n > 0) res *= n--; return res; }     static bool WTisPrimeSA(int n) { return ((factorial(n - 1) + 1) % n) == 0; }     static BI[] facts;     static void initFacts(int n) {        facts = new BI[n]; facts[0] = facts[1] = 1;        for (int i = 1, j = 2; j < n; i = j++)            facts[j] = facts[i] * j; }     static bool WTisPrime(int n) { return ((facts[n - 1] + 1) % n) == 0; }  // end stand-alone     static void Main(string[] args) { st = DateTime.Now;        BI f = 1; for (int n = 2; lst.Count < max; f *= n++) if ((f + 1) % n == 0) lst.Add(n);        ShowOne(ms1, ref et1);        st = DateTime.Now; int lmt = lst.Last(); lst.Clear(); BitArray flags = new BitArray(lmt + 1);        for (int n = 2; n <= lmt; n+=n==2?1:2) if (!flags[n]) {                lst.Add(n); for (int k = n * n, n2=n<<1; k <= lmt; k += n2) flags[k] = true; }        ShowOne(ms2, ref et2);        WriteLine("{0} was {1:0.0} times slower than the {2}.", ms1, et1 / et2, ms2);       // stand-alone computation        WriteLine("\n" + ms1 + " stand-alone computation:");        WriteLine("factorial computed for each item");        st = DateTime.Now;        for (int x = lst[skp - 1]; x <= lst[max - 1]; x++) if (WTisPrimeSA(x)) Write("{0,4} ", x);        WriteLine(); WriteLine("\nTime taken: {0}ms\n", (DateTime.Now - st).TotalMilliseconds);         WriteLine("factorials precomputed up to highest item");        st = DateTime.Now; initFacts(lst[max - 1]);        for (int x = lst[skp - 1]; x <= lst[max - 1]; x++) if (WTisPrime(x)) Write("{0,4} ", x);        WriteLine(); WriteLine("\nTime taken: {0}ms\n", (DateTime.Now - st).TotalMilliseconds);    }}`
Output @ Tio.run:
```--- Wilson's theorem method ---

The first 120 primes are:
2   3   5   7   11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1,000th prime thru the 1,015th prime:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081

Time taken: 340.901ms

--- Sieve of Eratosthenes method ---

The first 120 primes are:
2   3   5   7   11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1,000th prime thru the 1,015th prime:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081

Time taken: 2.118ms

Wilson's theorem method was 161.0 times slower than the Sieve of Eratosthenes method.

Wilson's theorem method stand-alone computation:
factorial computed for each item
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081

Time taken: 11265.2768ms

factorials precomputed up to highest item
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081

Time taken: 177.7401ms

```

The "slow" factor may be different on different processors and programming environments. For example, on Tio.run, the "slow" factor is anywhere between 120 and 180 times slower. Slowness most likely caused by the sluggish BigInteger library usage. The SoE method, although quicker, does consume some memory (due to the flags BitArray). The Wilson's theorem method may consume considerable memory due to the large factorials (the f variable) when computing larger primes.

The Wilson's theorem method is better suited for computing single primes, as the SoE method causes one to compute all the primes up to the desired item. In this C# implementation, a running factorial is maintained to help the Wilson's theorem method be a little more efficient. The stand-alone results show that when having to compute a BigInteger factorial for every primality test, the performance drops off considerably more. The last performance figure illustrates that memoizing the factorials can help when calculating nearby prime numbers.

## C++

`#include <iomanip>#include <iostream> int factorial_mod(int n, int p) {    int f = 1;    for (; n > 0 && f != 0; --n)        f = (f * n) % p;    return f;} bool is_prime(int p) {    return p > 1 && factorial_mod(p - 1, p) == p - 1;} int main() {    std::cout << "  n | prime?\n------------\n";    std::cout << std::boolalpha;    for (int p : {2, 3, 9, 15, 29, 37, 47, 57, 67, 77, 87, 97, 237, 409, 659})        std::cout << std::setw(3) << p << " | " << is_prime(p) << '\n';     std::cout << "\nFirst 120 primes by Wilson's theorem:\n";    int n = 0, p = 1;    for (; n < 120; ++p) {        if (is_prime(p))            std::cout << std::setw(3) << p << (++n % 20 == 0 ? '\n' : ' ');    }     std::cout << "\n1000th through 1015th primes:\n";    for (int i = 0; n < 1015; ++p) {        if (is_prime(p)) {            if (++n >= 1000)                std::cout << std::setw(4) << p << (++i % 16 == 0 ? '\n' : ' ');        }    }}`
Output:
```  n | prime?
------------
2 | true
3 | true
9 | false
15 | false
29 | true
37 | true
47 | true
57 | false
67 | true
77 | false
87 | false
97 | true
237 | false
409 | true
659 | true

First 120 primes by Wilson's theorem:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1000th through 1015th primes:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081
```

## CLU

`% Wilson primality testwilson = proc (n: int) returns (bool)    if n<2 then return (false) end     fac_mod: int := 1    for i: int in int\$from_to(2, n-1) do        fac_mod := fac_mod * i // n     end     return (fac_mod + 1 = n)end wilson  % Print primes up to 100 using Wilson's theoremstart_up = proc ()    po: stream := stream\$primary_output()    for i: int in int\$from_to(1, 100) do        if wilson(i) then               stream\$puts(po, int\$unparse(i) || " ")        end    end    stream\$putl(po, "")end start_up `
Output:
`2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97`

## Common Lisp

` (defun factorial (n)  (if (< n 2) 1 (* n (factorial (1- n)))) )  (defun primep (n) "Primality test using Wilson's Theorem"  (unless (zerop n)    (zerop (mod (1+ (factorial (1- n))) n)) )) `
Output:
```;; Primes under 20:
(dotimes (i 20) (when (primep i) (print i)))

1
2
3
5
7
11
13
17
19 ```

## Cowgol

`include "cowgol.coh"; # Wilson primality testsub wilson(n: uint32): (out: uint8) is    out := 0;    if n >= 2 then        var facmod: uint32 := 1;        var ct := n - 1;        while ct > 0 loop            facmod := (facmod * ct) % n;            ct := ct - 1;        end loop;        if facmod + 1 == n then            out := 1;        end if;    end if;end sub; # Print primes up to 100 according to Wilsonvar i: uint32 := 1;while i < 100 loop    if wilson(i) == 1 then        print_i32(i);        print_char(' ');    end if;    i := i + 1;end loop;print_nl();`
Output:
`2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97`

## D

Translation of: Java
`import std.bigint;import std.stdio; BigInt fact(long n) {    BigInt f = 1;    for (int i = 2; i <= n; i++) {        f *= i;    }    return f;} bool isPrime(long p) {    if (p <= 1) {        return false;    }    return (fact(p - 1) + 1) % p == 0;} void main() {    writeln("Primes less than 100 testing by Wilson's Theorem");    foreach (i; 0 .. 101) {        if (isPrime(i)) {            write(i, ' ');        }    }    writeln;}`
Output:
```Primes less than 100 testing by Wilson's Theorem
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97```

## Erlang

` #! /usr/bin/escript isprime(N) when N < 2 -> false;isprime(N) when N band 1 =:= 0 -> N =:= 2;isprime(N) -> fac_mod(N - 1, N) =:= N - 1. fac_mod(N, M) -> fac_mod(N, M, 1).fac_mod(1, _, A) -> A;fac_mod(N, M, A) -> fac_mod(N - 1, M, A*N rem M). main(_) ->    io:format("The first few primes (via Wilson's theorem) are: ~n~p~n",               [[K || K <- lists:seq(1, 128), isprime(K)]]). `
Output:
```The first few primes (via Wilson's theorem) are:
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,
103,107,109,113,127]
```

## F#

` // Wilsons theorem. Nigel Galloway: August 11th., 2020let wP(n,g)=(n+1I)%g=0Ilet fN=Seq.unfold(fun(n,g)->Some((n,g),((n*g),(g+1I))))(1I,2I)|>Seq.filter wPfN|>Seq.take 120|>Seq.iter(fun(_,n)->printf "%A " n);printfn "\n"fN|>Seq.skip 999|>Seq.take 15|>Seq.iter(fun(_,n)->printf "%A " n);printfn ""`
Output:
```2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069
```

## Factor

Works with: Factor version 0.99 2020-08-14
`USING: formatting grouping io kernel lists lists.lazy mathmath.factorials math.functions prettyprint sequences ; : wilson ( n -- ? ) [ 1 - factorial 1 + ] [ divisor? ] bi ;: prime? ( n -- ? ) dup 2 < [ drop f ] [ wilson ] if ;: primes ( -- list ) 1 lfrom [ prime? ] lfilter ; "n    prime?\n---  -----" print{ 2 3 9 15 29 37 47 57 67 77 87 97 237 409 659 }[ dup prime? "%-3d  %u\n" printf ] each nl "First 120 primes via Wilson's theorem:" print120 primes ltake list>array 20 group simple-table. nl "1000th through 1015th primes:" print16 primes 999 [ cdr ] times ltake list>array[ pprint bl ] each nl`
Output:
```n    prime?
---  -----
2    t
3    t
9    f
15   f
29   t
37   t
47   t
57   f
67   t
77   f
87   f
97   t
237  f
409  t
659  t

First 120 primes via Wilson's theorem:
2   3   5   7   11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97  101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1000th through 1015th primes:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081
```

## Fermat

`Func Wilson(n) = if ((n-1)!+1)|n = 0 then 1 else 0 fi.;`

## Forth

` : fac-mod ( n m -- r )    >r 1 swap    begin dup 0> while        dup rot * [email protected] mod  swap 1-    repeat drop rdrop ; : ?prime ( n -- f )    dup 1- tuck swap fac-mod = ; : .primes ( n -- )    cr 2 ?do i ?prime if i . then loop ; `
Output:
```128 .primes
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127  ok
```

## FreeBASIC

`function wilson_prime( n as uinteger ) as boolean    dim as uinteger fct=1, i    for i = 2 to n-1        'because   (a mod n)*b = (ab mod n)        'it is not necessary to calculate the entire factorial        fct = (fct * i) mod n    next i    if fct = n-1 then return true else return falseend function for i as uinteger = 2 to 100    if wilson_prime(i) then print i,next i`
Output:

Primes below 100

```2             3             5             7             11
13            17            19            23            29
31            37            41            43            47
53            59            61            67            71
73            79            83            89            97```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

## GAP

`# find primes using Wilson's theorem:#    p is prime if ( ( p - 1 )! + 1 ) mod p = 0 isWilsonPrime := function( p )    local fModP, i;    fModP := 1;    for i in [ 2 .. p - 1 ] do fModP := fModP * i; fModP := fModP mod p; od;    return fModP = p - 1;end; # isWilsonPrime prime := [];for i in [ -4 .. 100 ] do if isWilsonPrime( i ) then Add( prime, i ); fi; od;Display( prime );`
Output:
```[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97 ]
```

## Go

Needless to say, Wilson's theorem is an extremely inefficient way of testing for primalty with 'big integer' arithmetic being needed to compute factorials greater than 20.

Presumably we're not allowed to make any trial divisions here except by the number two where all even positive integers, except two itself, are obviously composite.

`package main import (    "fmt"    "math/big") var (    zero = big.NewInt(0)    one  = big.NewInt(1)    prev = big.NewInt(factorial(20))) // Only usable for n <= 20.func factorial(n int64) int64 {    res := int64(1)    for k := n; k > 1; k-- {        res *= k    }    return res} // If memo == true, stores previous sequential// factorial calculation for odd n > 21.func wilson(n int64, memo bool) bool {    if n <= 1 || (n%2 == 0 && n != 2) {        return false    }    if n <= 21 {        return (factorial(n-1)+1)%n == 0    }    b := big.NewInt(n)    r := big.NewInt(0)    z := big.NewInt(0)    if !memo {        z.MulRange(2, n-1) // computes factorial from scratch    } else {        prev.Mul(prev, r.MulRange(n-2, n-1)) // uses previous calculation        z.Set(prev)    }    z.Add(z, one)    return r.Rem(z, b).Cmp(zero) == 0    } func main() {    numbers := []int64{2, 3, 9, 15, 29, 37, 47, 57, 67, 77, 87, 97, 237, 409, 659}    fmt.Println("  n  prime")    fmt.Println("---  -----")    for _, n := range numbers {        fmt.Printf("%3d  %t\n", n, wilson(n, false))    }     // sequential memoized calculation    fmt.Println("\nThe first 120 prime numbers are:")    for i, count := int64(2), 0; count < 1015; i += 2 {        if wilson(i, true) {            count++            if count <= 120 {                fmt.Printf("%3d ", i)                if count%20 == 0 {                    fmt.Println()                }            } else if count >= 1000 {                if count == 1000 {                    fmt.Println("\nThe 1,000th to 1,015th prime numbers are:")                 }                fmt.Printf("%4d ", i)            }                    }        if i == 2 {            i--        }    }    fmt.Println()    }`
Output:
```  n  prime
---  -----
2  true
3  true
9  false
15  false
29  true
37  true
47  true
57  false
67  true
77  false
87  false
97  true
237  false
409  true
659  true

The first 120 prime numbers are:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

The 1,000th to 1,015th prime numbers are:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081
```

`import qualified Data.Text as Timport Data.List main = do    putStrLn \$ showTable True ' ' '-' ' ' \$ ["p","isPrime"]:map (\p -> [show p, show \$ isPrime p]) numbers    putStrLn \$ "The first 120 prime numbers are:"    putStrLn \$ see 20 \$ take 120 primes    putStrLn "The 1,000th to 1,015th prime numbers are:"    putStrLn \$ see 16.take 16 \$ drop 999 primes  numbers = [2,3,9,15,29,37,47,57,67,77,87,97,237,409,659] primes = [p | p <- 2:[3,5..], isPrime p] isPrime :: Integer -> BoolisPrime p = if p < 2 then False else 0 == mod (succ \$ product [1..pred p]) p bagOf :: Int -> [a] -> [[a]]bagOf _ [] = []bagOf n xs = let (us,vs) = splitAt n xs in us : bagOf n vs see :: Show a => Int -> [a] -> Stringsee n = unlines.map unwords.bagOf n.map (T.unpack.T.justifyRight 3 ' '.T.pack.show) showTable::Bool -> Char -> Char -> Char -> [[String]] -> StringshowTable _ _ _ _ [] = []showTable header ver hor sep contents = unlines \$ hr:(if header then z:hr:zs else intersperse hr zss) ++ [hr]   where   vss = map (map length) \$ contents   ms = map maximum \$ transpose vss ::[Int]   hr = concatMap (\ n -> sep : replicate n hor) ms ++ [sep]   top = replicate (length hr) hor   bss = map (\ps -> map (flip replicate ' ') \$ zipWith (-) ms ps) \$ vss   zss@(z:zs) = zipWith (\us bs -> (concat \$ zipWith (\x y -> (ver:x) ++ y) us bs) ++ [ver]) contents bss`
Output:
``` --- -------
p   isPrime
--- -------
2   True
3   True
9   False
15  False
29  True
37  True
47  True
57  False
67  True
77  False
87  False
97  True
237 False
409 True
659 True
--- -------

The first 120 prime numbers are:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

The 1,000th to 1,015th prime numbers are:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081
```

## J

`    wilson=: 0 = (| !&.:<:)   (#~ wilson) x: 2 + i. 302 3 5 7 11 13 17 19 23 29 31 `

## Java

Wilson's theorem is an extremely inefficient way of testing for primality. As a result, optimizations such as caching factorials not performed.

` import java.math.BigInteger; public class PrimaltyByWilsonsTheorem {     public static void main(String[] args) {        System.out.printf("Primes less than 100 testing by Wilson's Theorem%n");        for ( int i = 0 ; i <= 100 ; i++ ) {            if ( isPrime(i) ) {                System.out.printf("%d ", i);            }        }    }      private static boolean isPrime(long p) {        if ( p <= 1) {            return false;        }        return fact(p-1).add(BigInteger.ONE).mod(BigInteger.valueOf(p)).compareTo(BigInteger.ZERO) == 0;    }     private static BigInteger fact(long n) {        BigInteger fact = BigInteger.ONE;        for ( int i = 2 ; i <= n ; i++ ) {            fact = fact.multiply(BigInteger.valueOf(i));        }        return fact;    } } `
Output:
```Primes less than 100 testing by Wilson's Theorem
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## jq

Works with jq, subject to the limitations of IEEE 754 64-bit arithmetic.

Works with gojq, which supports unlimited-precision integer arithmetic.

`## Compute (n - 1)! mod m.def facmod(\$n; \$m):  reduce range(2; \$n+1) as \$k (1; (. * \$k) % \$m); def isPrime: .>1 and (facmod(. - 1; .) + 1) % . == 0; "Prime numbers between 2 and 100:",[range(2;101) | select (isPrime)], # Notice that `infinite` can be used as the second argument of `range`:"First 10 primes after 7900:",[limit(10; range(7900; infinite) | select(isPrime))]`
Output:
` Prime numbers between 2 and 100:[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]First 10 primes after 7900:[7901,7907,7919,7927,7933,7937,7949,7951,7963,7993]`

## Julia

`iswilsonprime(p) = (p < 2 || (p > 2 && iseven(p))) ? false : foldr((x, y) -> (x * y) % p, 1:p - 1) == p - 1 wilsonprimesbetween(n, m) = [i for i in n:m if iswilsonprime(i)] println("First 120 Wilson primes: ", wilsonprimesbetween(1, 1000)[1:120])println("\nThe first 40 Wilson primes above 7900 are: ", wilsonprimesbetween(7900, 9000)[1:40]) `
Output:
```First 120 Wilson primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659]

The first 40 Wilson primes above 7900 are: [7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269]
```

## Lua

`-- primality by Wilson's theorem function isWilsonPrime( n )    local fmodp = 1    for i = 2, n - 1 do        fmodp = fmodp * i        fmodp = fmodp % n    end    return fmodp == n - 1end for n = -1, 100 do    if isWilsonPrime( n ) then       io.write( " " .. n )    endend`
Output:
``` 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## Mathematica/Wolfram Language

`ClearAll[WilsonPrimeQ]WilsonPrimeQ[1] = False;WilsonPrimeQ[p_Integer] := Divisible[(p - 1)! + 1, p]Select[Range[100], WilsonPrimeQ]`
Output:

Prime factors up to a 100:

`{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}`

## Nim

`import strutils, sugar proc facmod(n, m: int): int =  ## Compute (n - 1)! mod m.  result = 1  for k in 2..n:    result = (result * k) mod m func isPrime(n: int): bool = (facmod(n - 1, n) + 1) mod n == 0 let primes = collect(newSeq):               for n in 2..100:                 if n.isPrime: n echo "Prime numbers between 2 and 100:"echo primes.join(" ")`
Output:
```Prime numbers between 2 and 100:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97```

## PARI/GP

`Wilson(n) = prod(i=1,n-1,Mod(i,n))==-1 `

## Perl

Library: ntheory
`use strict;use warnings;use feature 'say';use ntheory qw(factorial); my(\$ends_in_7, \$ends_in_3); sub is_wilson_prime {    my(\$n) = @_;    \$n > 1 or return 0;    (factorial(\$n-1) % \$n) == (\$n-1) ? 1 : 0;} for (0..50) {    my \$m = 3 + 10 * \$_;    \$ends_in_3 .= "\$m " if is_wilson_prime(\$m);    my \$n = 7 + 10 * \$_;    \$ends_in_7 .= "\$n " if is_wilson_prime(\$n);} say \$ends_in_3;say \$ends_in_7;`
Output:
```3 13 23 43 53 73 83 103 113 163 173 193 223 233 263 283 293 313 353 373 383 433 443 463 503
7 17 37 47 67 97 107 127 137 157 167 197 227 257 277 307 317 337 347 367 397 457 467 487```

## Phix

Uses the modulus method to avoid needing gmp, which was in fact about 7 times slower (when calculating the full factorials).

```function wilson(integer n)
integer facmod = 1
for i=2 to n-1 do
facmod = remainder(facmod*i,n)
end for
return facmod+1=n
end function

atom t0 = time()
sequence primes = {}
integer p = 2
while length(primes)<1015 do
if wilson(p) then
primes &= p
end if
p += 1
end while
printf(1,"The first 25 primes: %V\n",{primes[1..25]})
printf(1,"          builtin: %V\n",{get_primes(-25)})
printf(1,"primes[1000..1015]: %V\n",{primes[1000..1015]})
printf(1,"         builtin: %V\n",{get_primes(-1015)[1000..1015]})
?elapsed(time()-t0)
```
Output:
```The first 25 primes: {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}
'' builtin: {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}
primes[1000..1015]: {7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081}
'' builtin: {7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081}
"0.5s"
```

## Plain English

`To run:Start up.Show some primes (via Wilson's theorem).Wait for the escape key.Shut down. The maximum representable factorial is a number equal to 12.   \32-bit signed To show some primes (via Wilson's theorem):If a counter is past the maximum representable factorial, exit.If the counter is prime (via Wilson's theorem), write "" then the counter then " " on the console without advancing.Repeat. A prime is a number. A factorial is a number. To find a factorial of a number:Put 1 into the factorial.Loop.If a counter is past the number, exit.Multiply the factorial by the counter.Repeat. To decide if a number is prime (via Wilson's theorem):If the number is less than 1, say no.Find a factorial of the number minus 1. Bump the factorial.If the factorial is evenly divisible by the number, say yes.Say no.`
Output:
```1 2 3 5 7 11
```

## PL/I

`/* primality by Wilson's theorem */wilson: procedure options( main );   declare n binary(15)fixed;    isWilsonPrime: procedure( n )returns( bit(1) );      declare n            binary(15)fixed;      declare ( fmodp, i ) binary(15)fixed;      fmodp = 1;      do i = 2 to n - 1;         fmodp = mod( fmodp * i, n );      end;      return ( fmodp = n - 1 );   end isWilsonPrime ;    do n = 1 to 100;      if isWilsonPrime( n ) then do;         put edit( n ) ( f(3) );      end;   end;end wilson ;`
Output:
```  2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## PL/M

Works with: 8080 PL/M Compiler
... under CP/M (or an emulator)
`100H: /* FIND PRIMES USING WILSON'S THEOREM:                                */      /*      P IS PRIME IF ( ( P - 1 )! + 1 ) MOD P = 0                    */    DECLARE FALSE LITERALLY '0';    BDOS: PROCEDURE( FN, ARG ); /* CP/M BDOS SYSTEM CALL */      DECLARE FN BYTE, ARG ADDRESS;      GOTO 5;   END BDOS;   PRINT\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C ); END;   PRINT\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;   PRINT\$NUMBER: PROCEDURE( N );      DECLARE N ADDRESS;      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 PRINT\$STRING( .N\$STR( W ) );   END PRINT\$NUMBER;    /* RETURNS TRUE IF P IS PRIME BY WILSON'S THEOREM, FALSE OTHERWISE       */   /*         COMPUTES THE FACTORIAL MOD P AT EACH STAGE, SO AS TO ALLOW    */   /*         FOR NUMBERS WHOSE FACTORIAL WON'T FIT IN 16 BITS              */   IS\$WILSON\$PRIME: PROCEDURE( P )BYTE;      DECLARE P ADDRESS;      IF P < 2 THEN RETURN FALSE;      ELSE DO;         DECLARE ( I, FACTORIAL\$MOD\$P ) ADDRESS;         FACTORIAL\$MOD\$P = 1;         DO I = 2 TO P - 1;            FACTORIAL\$MOD\$P = ( FACTORIAL\$MOD\$P * I ) MOD P;         END;         RETURN FACTORIAL\$MOD\$P = P - 1;      END;   END IS\$WILSON\$PRIME;    DECLARE I ADDRESS;   DO I = 1 TO 100;      IF IS\$WILSON\$PRIME( I ) THEN DO;         CALL PRINT\$CHAR( ' ' );         CALL PRINT\$NUMBER( I );      END;   END; EOF`
Output:
``` 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## Polyglot:PL/I and PL/M

The following Primality by Wilson's theorem solution will run under both PL/M and PL/I.

Works with: 8080 PL/M Compiler
... under CP/M (or an emulator)

Should work with many PL/I implementations.
The PL/I include file "pg.inc" can be found on the Polyglot:PL/I and PL/M page. Note the use of text in column 81 onwards to hide the PL/I specifics from the PL/M compiler.

`/* PRIMALITY BY WILSON'S THEOREM */wilson_100H: procedure options                                                  (main); /* PL/I DEFINITIONS                                                             */%include 'pg.inc';/* PL/M DEFINITIONS: CP/M BDOS SYSTEM CALL AND CONSOLE I/O ROUTINES, ETC. */    /*   DECLARE BINARY LITERALLY 'ADDRESS', CHARACTER LITERALLY 'BYTE';   DECLARE SADDR  LITERALLY '.',       BIT       LITERALLY 'BYTE';   BDOSF: PROCEDURE( FN, ARG )BYTE;                               DECLARE FN BYTE, ARG ADDRESS; GOTO 5;   END;    BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5;   END;   PRCHAR:   PROCEDURE( C );   DECLARE C CHARACTER; CALL BDOS( 2, C ); END;   PRNL:     PROCEDURE;        CALL PRCHAR( 0DH ); CALL PRCHAR( 0AH ); END;   PRNUMBER: PROCEDURE( N );      DECLARE N ADDRESS;      DECLARE V ADDRESS, N\$STR( 6 ) BYTE, W BYTE;      N\$STR( W := LAST( N\$STR ) ) = '\$';      N\$STR( W := W - 1 ) = '0' + ( ( V := N ) MOD 10 );      DO WHILE( ( V := V / 10 ) > 0 );         N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );      END;       CALL BDOS( 9, .N\$STR( W ) );   END PRNUMBER;   MODF: PROCEDURE( A, B )ADDRESS;      DECLARE ( A, B )ADDRESS;      RETURN( A MOD B );   END MODF;/* END LANGUAGE DEFINITIONS */    /* TASK */   DECLARE N BINARY;    ISWILSONPRIME: PROCEDURE( N )returns                                         (                                BIT                                             )                                ;      DECLARE N            BINARY;      DECLARE ( FMODP, I ) BINARY;      FMODP = 1;      DO I = 2 TO N - 1;         FMODP = MODF( FMODP * I, N );      END;      RETURN ( FMODP = N - 1 );   END ISWILSONPRIME ;    DO N = 1 TO 100;      IF ISWILSONPRIME( N ) THEN DO;         CALL PRCHAR( ' ' );         CALL PRNUMBER( N );      END;   END; EOF: end wilson_100H ;`
Output:
``` 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## Python

No attempt is made to optimise this as this method is a very poor primality test.

`from math import factorial def is_wprime(n):    return n > 1 and bool(n == 2 or                          (n % 2 and (factorial(n - 1) + 1) % n == 0)) if __name__ == '__main__':    c = 100    print(f"Primes under {c}:", end='\n  ')    print([n for n in range(c) if is_wprime(n)])`
Output:
```Primes under 100:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]```

## Quackery

` [ 1 swap times [ i 1+ * ] ] is !     ( n --> n )  [ dup 2 < iff     [ drop false ] done    dup 1 - ! 1+   swap mod 0 = ]            is prime ( n --> b )  say "Primes less than 500: " 500 times    [ i^ prime if        [ i^ echo sp ] ]`
Output:
`Primes less than 500: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 `

## Raku

(formerly Perl 6)

Works with: Rakudo version 2019.11

Not a particularly recommended way to test for primality, especially for larger numbers. It works, but is slow and memory intensive.

`sub postfix:<!> (Int \$n) { (constant f = 1, |[\*] 1..*)[\$n] } sub is-wilson-prime (Int \$p where * > 1) { ((\$p - 1)! + 1) %% \$p } # Pre initialize factorial routine (not thread safe)9000!; # Testingput '   p  prime?';printf("%4d  %s\n", \$_, .&is-wilson-prime) for 2, 3, 9, 15, 29, 37, 47, 57, 67, 77, 87, 97, 237, 409, 659; my \$wilsons = (2,3,*+2…*).hyper.grep: &is-wilson-prime; put "\nFirst 120 primes:";put \$wilsons[^120].rotor(20)».fmt('%3d').join: "\n"; put "\n1000th through 1015th primes:";put \$wilsons[999..1014];`
Output:
```   p  prime?
2  True
3  True
9  False
15  False
29  True
37  True
47  True
57  False
67  True
77  False
87  False
97  True
237  False
409  True
659  True

First 120 primes:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1000th through 1015th primes:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081```

## REXX

Some effort was made to optimize the factorial computation by using memoization and also minimize the size of the
decimal digit precision     (NUMERIC DIGITS expression).

Also, a "pretty print" was used to align the displaying of a list.

`/*REXX pgm tests for primality via Wilson's theorem: a # is prime if p divides (p-1)! +1*/parse arg LO zz                                  /*obtain optional arguments from the CL*/if LO=='' | LO==","  then LO= 120                /*Not specified?  Then use the default.*/if zz ='' | zz =","  then zz=2 3 9 15 29 37 47 57 67 77 87 97 237 409 659 /*use default?*/sw= linesize() - 1;  if sw<1  then sw= 79        /*obtain the terminal's screen width.  */digs = digits()                                  /*the current number of decimal digits.*/#= 0                                             /*number of  (LO)  primes found so far.*/!.= 1                                            /*placeholder for factorial memoization*/\$=                                               /*     "      to hold a list of primes.*/    do p=1  until #=LO;         oDigs= digs      /*remember the number of decimal digits*/    ?= isPrimeW(p)                               /*test primality using Wilson's theorem*/    if digs>Odigs  then numeric digits digs      /*use larger number for decimal digits?*/    if \?  then iterate                          /*if not prime, then ignore this number*/    #= # + 1;                   \$= \$ p           /*bump prime counter; add prime to list*/    end   /*p*/ call show 'The first '    LO    " prime numbers are:"w= max( length(LO), length(word(reverse(zz),1))) /*used to align the number being tested*/@is.0= "            isn't";     @is.1= 'is'      /*2 literals used for display: is/ain't*/say    do z=1  for words(zz);      oDigs= digs      /*remember the number of decimal digits*/    p= word(zz, z)                               /*get a number from user─supplied list.*/    ?= isPrimeW(p)                               /*test primality using Wilson's theorem*/    if digs>Odigs  then numeric digits digs      /*use larger number for decimal digits?*/    say right(p, max(w,length(p) ) )       @is.?      "prime."    end   /*z*/exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/isPrimeW: procedure expose !. digs;  parse arg x '' -1 last;        != 1;       xm= x - 1          if x<2                   then return 0 /*is the number too small to be prime? */          if x==2 | x==5           then return 1 /*is the number a two or a five?       */          if last//2==0 | last==5  then return 0 /*is the last decimal digit even or 5? */          if !.xm\==1  then != !.xm              /*has the factorial been pre─computed? */                       else do;  if xm>!.0  then do; base= !.0+1; _= !.0;  != !._; end                                            else     base= 2        /* [↑] use shortcut.*/                                      do j=!.0+1  to xm;  != ! * j  /*compute factorial.*/                                      if pos(., !)\==0  then do;  parse var !  'E'  expon                                                                  numeric digits expon +99                                                                  digs = digits()                                                             end    /* [↑] has exponent,*/                                      end   /*j*/                   /*bump numeric digs.*/                            if xm<999  then do; !.xm=!; !.0=xm; end /*assign factorial. */                            end                                     /*only save small #s*/          if (!+1)//x==0  then return 1                             /*X  is     a prime.*/                               return 0                             /*"  isn't  "   "   *//*──────────────────────────────────────────────────────────────────────────────────────*/show: parse arg header,oo;     say header        /*display header for the first N primes*/      w= length( word(\$, LO) )                   /*used to align prime numbers in \$ list*/        do k=1  for LO; _= right( word(\$, k), w) /*build list for displaying the primes.*/        if length(oo _)>sw  then do;  say substr(oo,2);  oo=;  end  /*a line overflowed?*/        oo= oo _                                                    /*display a line.   */        end   /*k*/                                                 /*does pretty print.*/      if oo\=''  then say substr(oo, 2);  return /*display residual (if any overflowed).*/`

Programming note:   This REXX program makes use of   LINESIZE   REXX program   (or BIF)   which is used to determine the screen width
(or linesize)   of the terminal (console).   Some REXXes don't have this BIF.

The   LINESIZE.REX   REXX program is included here   ───►   LINESIZE.REX.

output   when using the default inputs:
```The first  120  prime numbers are:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

2 is prime.
3 is prime.
9             isn't prime.
15             isn't prime.
29 is prime.
37 is prime.
47 is prime.
57             isn't prime.
67 is prime.
77             isn't prime.
87             isn't prime.
97 is prime.
237             isn't prime.
409 is prime.
659 is prime.
```

## Ring

` load "stdlib.ring" decimals(0)limit = 19 for n = 2 to limit    fact = factorial(n-1) + 1    see "Is " + n + " prime: "    if fact % n = 0       see "1" + nl    else       see "0" + nl    oknext `

Output:

```Is 2 prime: 1
Is 3 prime: 1
Is 4 prime: 0
Is 5 prime: 1
Is 6 prime: 0
Is 7 prime: 1
Is 8 prime: 0
Is 9 prime: 0
Is 10 prime: 0
Is 11 prime: 1
Is 12 prime: 0
Is 13 prime: 1
Is 14 prime: 0
Is 15 prime: 0
Is 16 prime: 0
Is 17 prime: 1
Is 18 prime: 0
Is 19 prime: 1
```

Alternative version computing the factorials modulo n so as to avoid overflow.

`# primality by Wilson's theorem limit = 100 for n = 1 to limit    if isWilsonPrime( n )       see " " + n    oknext n func isWilsonPrime n    fmodp = 1    for i = 2 to n - 1        fmodp *= i        fmodp %= n    next i    return fmodp = n - 1`
Output:
``` 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

## Ruby

`def w_prime?(i)  return false if i < 2  ((1..i-1).inject(&:*) + 1) % i == 0end p (1..100).select{|n| w_prime?(n) } `
Output:
```[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```

## Rust

`fn factorial_mod(mut n: u32, p: u32) -> u32 {    let mut f = 1;    while n != 0 && f != 0 {        f = (f * n) % p;        n -= 1;    }    f} fn is_prime(p: u32) -> bool {    p > 1 && factorial_mod(p - 1, p) == p - 1} fn main() {    println!("  n | prime?\n------------");    for p in vec![2, 3, 9, 15, 29, 37, 47, 57, 67, 77, 87, 97, 237, 409, 659] {        println!("{:>3} | {}", p, is_prime(p));    }    println!("\nFirst 120 primes by Wilson's theorem:");    let mut n = 0;    let mut p = 1;    while n < 120 {        if is_prime(p) {            n += 1;            print!("{:>3}{}", p, if n % 20 == 0 { '\n' } else { ' ' });        }        p += 1;    }    println!("\n1000th through 1015th primes:");    let mut i = 0;    while n < 1015 {        if is_prime(p) {            n += 1;            if n >= 1000 {                i += 1;                print!("{:>3}{}", p, if i % 16 == 0 { '\n' } else { ' ' });            }        }        p += 1;    }}`
Output:
```  n | prime?
------------
2 | true
3 | true
9 | false
15 | false
29 | true
37 | true
47 | true
57 | false
67 | true
77 | false
87 | false
97 | true
237 | false
409 | true
659 | true

First 120 primes by Wilson's theorem:
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71
73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659

1000th through 1015th primes:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081
```

## Sidef

`func is_wilson_prime_slow(n) {    n > 1 || return false    (n-1)! % n == n-1} func is_wilson_prime_fast(n) {    n > 1 || return false    factorialmod(n-1, n) == n-1} say 25.by(is_wilson_prime_slow)     #=> [2, 3, 5, ..., 83, 89, 97]say 25.by(is_wilson_prime_fast)     #=> [2, 3, 5, ..., 83, 89, 97] say is_wilson_prime_fast(2**43 - 1)   #=> falsesay is_wilson_prime_fast(2**61 - 1)   #=> true`

## Swift

Using a BigInt library.

`import BigInt func factorial<T: BinaryInteger>(_ n: T) -> T {  guard n != 0 else {    return 1  }   return stride(from: n, to: 0, by: -1).reduce(1, *)}  func isWilsonPrime<T: BinaryInteger>(_ n: T) -> Bool {  guard n >= 2 else {    return false  }   return (factorial(n - 1) + 1) % n == 0} print((1...100).map({ BigInt(\$0) }).filter(isWilsonPrime))`
Output:
`[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]`

## Tiny BASIC

`    PRINT "Number to test"    INPUT N    IF N < 0 THEN LET N = -N    IF N = 2 THEN GOTO 30    IF N < 2 THEN GOTO 40     LET F = 1    LET J = 110  LET J = J + 1    REM exploits the fact that (F mod N)*J = (F*J mod N)    REM to do the factorial without overflowing    LET F = F * J    GOSUB 20    IF J  < N - 1 THEN GOTO 10    IF F  = N - 1 THEN PRINT "It is prime"    IF F <> N - 1 THEN PRINT "It is not prime"    END20  REM modulo by repeated subtraction    IF F < N THEN RETURN    LET F = F - N    GOTO 2030  REM special case N=2    PRINT "It is prime"    END40  REM zero and one are nonprimes by definition    PRINT "It is not prime"    END`

## Wren

Library: Wren-math
Library: Wren-fmt

Due to a limitation in the size of integers which Wren can handle (2^53-1) and lack of big integer support, we can only reliably demonstrate primality using Wilson's theorem for numbers up to 19.

`import "/math" for Intimport "/fmt" for Fmt var wilson = Fn.new { |p|    if (p < 2) return false    return (Int.factorial(p-1) + 1) % p == 0} for (p in 1..19) {    Fmt.print("\$2d -> \$s", p, wilson.call(p) ? "prime" : "not prime")}`
Output:
``` 1 -> not prime
2 -> prime
3 -> prime
4 -> not prime
5 -> prime
6 -> not prime
7 -> prime
8 -> not prime
9 -> not prime
10 -> not prime
11 -> prime
12 -> not prime
13 -> prime
14 -> not prime
15 -> not prime
16 -> not prime
17 -> prime
18 -> not prime
19 -> prime
```

## zkl

Library: GMP
GNU Multiple Precision Arithmetic Library and primes
`var [const] BI=Import("zklBigNum");  // libGMPfcn isWilsonPrime(p){   if(p<=1 or (p%2==0 and p!=2)) return(False);   BI(p-1).factorial().add(1).mod(p) == 0}fcn wPrimesW{ [2..].tweak(fcn(n){ isWilsonPrime(n) and n or Void.Skip }) }`
`numbers:=T(2, 3, 9, 15, 29, 37, 47, 57, 67, 77, 87, 97, 237, 409, 659);println("  n  prime");println("---  -----");foreach n in (numbers){ println("%3d  %s".fmt(n, isWilsonPrime(n))) } println("\nFirst 120 primes via Wilson's theorem:");wPrimesW().walk(120).pump(Void, T(Void.Read,15,False),   fcn(ns){ vm.arglist.apply("%4d".fmt).concat(" ").println() }); println("\nThe 1,000th to 1,015th prime numbers are:");wPrimesW().drop(999).walk(15).concat(" ").println();`
Output:
```  n  prime
---  -----
2  True
3  True
9  False
15  False
29  True
37  True
47  True
57  False
67  True
77  False
87  False
97  True
237  False
409  True
659  True

First 120 primes via Wilson's theorem:
2    3    5    7   11   13   17   19   23   29   31   37   41   43   47   53
59   61   67   71   73   79   83   89   97  101  103  107  109  113  127  131
137  139  149  151  157  163  167  173  179  181  191  193  197  199  211  223
227  229  233  239  241  251  257  263  269  271  277  281  283  293  307  311
313  317  331  337  347  349  353  359  367  373  379  383  389  397  401  409
419  421  431  433  439  443  449  457  461  463  467  479  487  491  499  503
509  521  523  541  547  557  563  569  571  577  587  593  599  601  607  613
617  619  631  641  643  647  653  659

The 1,000th to 1,015th prime numbers are:
7919 7927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069
```