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
 Primality by Wilson's theorem


#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)))