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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]].\nThen $(p - 1)! \\equiv -1 \\pmod p$.\n\n## Remarks\n\nA 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.\nFor example, $4! = 24 \\equiv -1 \\pmod 5$ and $6! = 720 \\equiv -1 \\pmod 7$.\nThe [[converse of Wilson's theorem|06WRV8HFDY]] also holds, so the congruence characterizes the primes.\n\n## History\n\nIbn 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.\n\n## Explanations\n\n### L1 human\n\nMultiply all the numbers from $1$ to $p - 1$, and divide by $p$. The remainder is always $p - 1$.\nFor $p = 7$: $1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 = 720 = 102 \\cdot 7 + 6$.\n",
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