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?

  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – draks ... Dec 20 '12 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. $\endgroup$ – André Nicolas Dec 20 '12 at 18:21

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


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