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  "body": "\n## Definition\n\nA Carmichael number is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \\equiv 1 \\pmod n$ for each integer $a$ [[coprime|0GFKP8F8AG]] to $n$.\n\n## Remarks\n\nThus a Carmichael number is a [[Fermat pseudoprime|02M2K4QH9P]] to each base coprime to it.\nKorselt's criterion: a composite $n$ is a Carmichael number if and only if $n$ is square-free and $p - 1 \\mid n - 1$ for each prime $p \\mid n$.\nThe smallest Carmichael number is $561 = 3 \\cdot 11 \\cdot 17$. Here $2$, $10$ and $16$ divide $560$.\nAlford, Granville and Pomerance proved in 1994 that there are infinitely many Carmichael numbers.\n\n## Explanations\n\n### L1 human\n\nA Carmichael number behaves like a prime for the Fermat test, for each base that has no common factor with it.\nSo the [[Fermat primality test|0N5ZSZMYH6]] cannot detect it, unless the base shares a factor with it.\n",
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