In mathematics, a Wilson prime is a certain kind of prime number. A prime p is called a Wilson prime if 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.
The only known Wilson primes are 5, 13, and 563 (Sloane's A007540); if any others exist, they must be > 5 · 108. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y) / log(x)).
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