2
$\begingroup$

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?

  • 0
    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.2012-12-20

1 Answers 1

5

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