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Let $p$ be prime, and $a$ be integer. When does $(p - 1)! + 1 = p^a$ for some $a$ hold?

For example: $p = 5 \implies (5 - 1)! + 1 = 25 = 5^2$ $p = 7 \implies (7 - 1)! + 1 = 721 = 7 \cdot 103$

Any idea?

Thanks,

  • 0
    Apparently $13^2|\ ((13-1)!+1)$ as well.2011-03-13

1 Answers 1

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A Wilson prime is a prime number $p$ such that $p^2$ divides $(p-1)!+1$. The only known Wilson primes are 5, 13, and 563.

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    Thanks. A big surprise for me though.2011-03-13