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Wilson's theorem states that a natural number $n>1$ is a prime number if and only if

$$ (n-1)! \equiv -1 \pmod {n} $$

Can we prove it using Fermat's Little theorem? If yes, then how?

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  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – draks ...
    Dec 20, 2012 at 16:42
  • $\begingroup$ Proving that if $n\gt 1$ and $(n-1)!\equiv -1\pmod{n}$ then $n$ is prime will not involve Fermat's Theorem, but that is the easy direction. $\endgroup$ Dec 20, 2012 at 18:21

1 Answer 1

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Hint: Consider $(x-1)(x-2)...(x-(p-1))$.

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