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  "body": "\n## Summary\n\nAn odd [[prime number|01826KMG2H]] $p$ is regular if it does not divide the [[class number|KTKK9B1X41]] of the [[cyclotomic field|NN77VYSGXR]] $\\Q(\\zeta_p)$.\nKummer introduced regular primes and proved [[Fermat's Last Theorem|0Y5X3SDBCZ]] for every regular prime exponent.\nBy Kummer's criterion, $p$ is regular exactly when $p$ divides none of the numerators of the [[Bernoulli numbers|0KC00K7KY4]] $B_2, B_4, \\ldots, B_{p-3}$.\nThe smallest irregular prime is $37$.\n\n## Explanations\n\n### L1 human\n\nKummer tried to prove Fermat's Last Theorem by factoring $x^p + y^p$ in the ring generated by a $p$-th root of unity.\nThat ring does not always have unique factorisation, and the class number measures the failure.\nWhen $p$ does not divide this class number, the failure is mild enough for his argument to work; such primes are called regular.\nAll odd primes below $37$ are regular, while $37$ is not, because $37$ divides the numerator of $B_{32}$.\n",
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