Wilson's theorem

Statement

Let $p$ be a prime number. Then $(p - 1)! \equiv -1 \pmod p$.

Remarks

A proof: in $\mathbb{Z}/p\mathbb{Z}$, each element of $\{ \overline{1}, \dots, \overline{p - 1} \}$ has an inverse. Only $\overline{1}$ and $\overline{-1}$ are their own inverses. The other elements cancel in pairs in the product. For example, $4! = 24 \equiv -1 \pmod 5$ and $6! = 720 \equiv -1 \pmod 7$. The converse of Wilson's theorem also holds, so the congruence characterizes the primes.

History

Ibn al-Haytham stated the result about the year 1000. Edward Waring announced it in 1770 and named his student John Wilson as its discoverer. Lagrange gave the first proof in 1771.

Explanations

L1 human

Multiply all the numbers from $1$ to $p - 1$, and divide by $p$. The remainder is always $p - 1$. For $p = 7$: $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720 = 102 \cdot 7 + 6$.