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  "body": "\n## Definition\n\nLet $p$ be an odd [[prime number|01826KMG2H]], and let $\\zeta_p$ be a primitive $p$-th [[root of unity|0YV2CKP2AV]].\nThe prime $p$ is *regular* if $p$ does not divide the [[class number|KTKK9B1X41]] of the [[cyclotomic field|NN77VYSGXR]] $\\Q(\\zeta_p)$.\nOtherwise $p$ is *irregular*.\n\n## Remarks\n\nKummer's criterion gives an equivalent form: $p$ is regular if and only if $p$ does not divide the numerator of any of the [[Bernoulli numbers|0KC00K7KY4]] $B_k$ for $k = 2, 4, \\ldots, p - 3$.\nKummer proved that if $p$ is regular, then $x^p + y^p = z^p$ has no solution in positive integers, the regular case of [[Fermat's Last Theorem|0Y5X3SDBCZ]].\nIt is known that there are infinitely many irregular primes.\nWhether there are infinitely many regular primes is an open question.\n\nExample: $p = 5$ is regular.\nBy the criterion, only $B_2 = \\frac{1}{6}$ needs checking, and $5$ does not divide its numerator $1$; in fact $\\Q(\\zeta_5)$ has class number $1$.\n\nNon-example: $p = 37$ is irregular.\nThe numerator of $B_{32}$ is divisible by $37$, and $32 \\le 37 - 3$; equivalently, $37$ divides the class number of $\\Q(\\zeta_{37})$.\n",
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