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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer that $p$ does not divide.\nThen $a^{p-1} \\equiv 1 \\pmod p$.\n\n## Remarks\n\nAn equivalent form is [[Fermat's little theorem in the form a^p = a|0BMXWXDVJZ]]: $a^p \\equiv a \\pmod p$ for each integer $a$.\nIn the ring $\\mathbb{Z}/p\\mathbb{Z}$, the statement is $x^{p-1} = 1$ for each nonzero $x$. It follows from [[Lagrange's theorem|3C53TD]] for the multiplicative [[group|G5S1M9]] of $\\mathbb{Z}/p\\mathbb{Z}$, which has $p - 1$ elements.\n[[Euler's theorem|0VC4DVE7DQ]] generalizes the result to each modulus $n$.\nThe converse is false. For example, $2^{340} \\equiv 1 \\pmod{341}$, but $341 = 11 \\cdot 31$. See [[Fermat pseudoprime|02M2K4QH9P]].\n\n## History\n\nFermat stated the theorem in a letter of 1640, without a proof. Euler published the first known proof in 1736. Leibniz had a proof earlier, in an unpublished manuscript.\n\n## Explanations\n\n### L1 human\n\nTake $p = 5$ and $a = 2$: $2^4 = 16 = 3 \\cdot 5 + 1$, so $2^4 \\equiv 1 \\pmod 5$.\nThe theorem says that this always happens for a prime $p$ and a number $a$ that $p$ does not divide.\nIt lets you reduce large powers: $2^{100} = (2^4)^{25} \\equiv 1 \\pmod 5$.\n",
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