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Wilson prime

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Named after
  
John Wilson

Author of publication
  
First terms
  
5, 13, 563

Publication year
  
1938

No. of known terms
  
3

Largest known term
  
563

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

Contents

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS); if any others exist, they must be greater than 2×1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [xy] is about log(log(y)/log(x)).

Several computer searches have been done in the hope of finding new Wilson primes. The Ibercivis distributed computing project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.

Wilson primes of order n

Wilson's theorem can be expressed in general as ( n 1 ) ! ( p n ) ! ( 1 ) n   mod p for every integer n 1 and prime p n . Generalized Wilson primes of order n are the primes p such that p 2 divides ( n 1 ) ! ( p n ) ! ( 1 ) n .

It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n.

Least generalized Wilson prime of order n are

5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4×107) (sequence A128666 in the OEIS)

Near-Wilson primes

A prime p satisfying the congruence (p − 1)! ≡ − 1 + Bp mod p2 with small | B | can be called a near-Wilson prime. Near-Wilson primes with B = 0 represent Wilson primes. The following table lists all such primes with | B | ≤ 100 from 106 up to 4×1011:

Wilson numbers

A Wilson number is a natural number n such that W(n) ≡ 0 (mod n2), where W ( n ) = gcd ( k , n ) = 1 1 k n k + e , the constant e = 1 if and only if n have a primitive root, otherwise, e = -1 For every natural number n, W(n) is divisible by n, and the quotients (called generalized Wilson quotients) are listed in  A157249. The Wilson numbers are

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in the OEIS)

If a Wilson number n is prime, then n is a Wilson prime. There are 13 Wilson numbers up to 5×108.

References

Wilson prime Wikipedia


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