Wilson primes of order n
Wilson primes of order n is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
- Definition
A Wilson prime of order n is a prime number p such that p² divides exactly:
(n − 1)! x (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
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) {
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