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.
- Task
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.
- Related task
ALGOL 68
which is
which is
Algol 68G supports long integers, however the precision must be specified.
As the memory required for a limit of 11 000 would exceed he maximum supported by Algol 68G under WIndows, a limit of 5 500 is used here, which is sufficient to find all but the 4th order Wilson prime.
<lang algol68>BEGIN # find Wilson primes of order n, primes such that: #
# ( ( n - 1 )! x ( p - n )! - (-1)^n ) mod p^2 = 0 #
INT limit = 5 508; # max prime to consider #
# Build list of primes. #
[]INT primes =
BEGIN
# sieve the primes to limit^2 which will hopefully be enough for primes #
[ 1 : limit * limit ]BOOL prime;
prime[ 1 ] := FALSE; prime[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO UPB prime DO prime[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO UPB prime DO prime[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( UPB prime ) DO
IF prime[ i ] THEN FOR s FROM i * i BY i + i TO UPB prime DO prime[ s ] := FALSE OD FI
OD;
# count the primes up to the limit #
INT p count := 0; FOR i TO limit DO IF prime[ i ] THEN p count +:= 1 FI OD;
# construct a list of the primes #
[ 1 : p count ]INT primes;
INT p pos := 0;
FOR i WHILE p pos < UPB primes DO IF prime[ i ] THEN primes[ p pos +:= 1 ] := i FI OD;
primes
END;
# Build list of factorials. # PR precision 20000 PR # set the number of digits for a LONG LONG INT # [ 0 : primes[ UPB primes ] ]LONG LONG INT facts; facts[ 0 ] := 1; FOR i TO UPB facts DO facts[ i ] := facts[ i - 1 ] * i OD;
# find the Wilson primes #
INT sign := 1;
print( ( " n: Wilson primes", newline ) );
print( ( "-----------------", newline ) );
FOR n TO 11 DO
print( ( whole( n, -2 ), ":" ) );
sign := - sign;
LONG LONG INT f n minus 1 = facts[ n - 1 ];
FOR p pos FROM LWB primes TO UPB primes DO
INT p = primes[ p pos ];
IF p >= n THEN
LONG LONG INT f = f n minus 1 * facts[ p - n ] - sign;
IF f MOD ( p * p ) = 0 THEN print( ( " ", whole( p, 0 ) ) ) FI
FI
OD;
print( ( newline ) )
OD
END</lang>
- Output:
n: Wilson primes ----------------- 1: 5 13 563 2: 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
C++
<lang cpp>#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';
}
}</lang>
- 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
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // 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 "") </lang>
- 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
<lang factor>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
$[ limit [1,b] 1 [ * ] accumulate* 1 prefix ]
- factorial ( n -- n! ) factorials nth ; inline
INFIX:: fn ( p n -- m )
factorial(n-1) * factorial(p-n) - -1**n ;
- wilson? ( p n -- ? ) [ fn ] keepd sq divisor? ; inline
- order ( n -- seq )
primes swap [ [ < ] curry drop-while ] keep [ wilson? ] curry filter ;
- order. ( n -- )
dup "%2d: " printf order [ pprint bl ] each nl ;
" n: Wilson primes\n--------------------" print 11 [1,b] [ order. ] each</lang>
- Output:
n: Wilson primes -------------------- 1: 5 13 563 2: 2 3 11 107 4931 3: 7 4: 10429 5: 5 7 47 6: 11 7: 17 8: 9: 541 10: 11 1109 11: 17 2713
Go
<lang go>package main
import (
"fmt" "math/big" "rcu"
)
func main() {
const LIMIT = 11000
primes := rcu.Primes(LIMIT)
facts := make([]*big.Int, LIMIT)
facts[0] = big.NewInt(1)
for i := int64(1); i < LIMIT; i++ {
facts[i] = new(big.Int)
facts[i].Mul(facts[i-1], big.NewInt(i))
}
sign := int64(1)
f := new(big.Int)
zero := new(big.Int)
fmt.Println(" n: Wilson primes")
fmt.Println("--------------------")
for n := 1; n < 12; n++ {
fmt.Printf("%2d: ", n)
sign = -sign
for _, p := range primes {
if p < n {
continue
}
f.Mul(facts[n-1], facts[p-n])
f.Sub(f, big.NewInt(sign))
p2 := int64(p * p)
bp2 := big.NewInt(p2)
if f.Rem(f, bp2).Cmp(zero) == 0 {
fmt.Printf("%d ", p)
}
}
fmt.Println()
}
}</lang>
- Output:
n: Wilson primes -------------------- 1: 5 13 563 2: 2 3 11 107 4931 3: 7 4: 10429 5: 5 7 47 6: 11 7: 17 8: 9: 541 10: 11 1109 11: 17 2713
Julia
<lang julia>using Primes
function wilsonprimes(limit = 11000)
sgn, facts = 1, accumulate(*, 1:limit, init = big"1")
println(" n: Wilson primes\n--------------------")
for n in 1:11
print(lpad(n, 2), ": ")
sgn = -sgn
for p in primes(limit)
if p > n && (facts[n < 2 ? 1 : n - 1] * facts[p - n] - sgn) % p^2 == 0
print("$p ")
end
end
println()
end
end
wilsonprimes() </lang> Output: Same as Wren example.
Mathematica/Wolfram Language
<lang Mathematica>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[out2, {}]
]
Do[
Print[WilsonPrime[n]]
,
{n, 1, 11}
]</lang>
- Output:
{5,13,563}
{2,3,11,107,4931}
{7}
{10429}
{5,7,47}
{11}
{17}
{}
{541}
{11,1109}
{17,2713}
Nim
As in Nim there is not (not yet?) a standard module to deal with big numbers, we use the third party module “bignum”. <lang Nim>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:
primes.add p.toInt 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:
wilson.add p
echo &"{n:2}: ", wilson.join(" ")</lang>
- 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
Perl
<lang perl>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
}</lang>
- 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
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.
Raku
<lang perl6># 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
}</lang>
- 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. <lang rexx>/*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</lang>
- 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 ───────┴─────────────────────────────────────────────────────────────
Rust
<lang 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;
}
}</lang>
- 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
<lang ruby>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) })
}</lang>
- 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
<lang ecmascript>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()
}</lang>
- 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