11.1. Background: irregular primes and Vandiver's Theorem III🔗

Kummer's criterion characterises regular primes: p is regular iff p does not divide the numerator of any of the Bernoulli numbers B_2, B_4, \ldots, B_{p-3}. Kummer showed in 1850 that \mathrm{FLT}_p holds for every regular prime p. This already proves \mathrm{FLT}_p for the vast majority of primes; the smallest irregular primes are 37, 59, 67, 101, 103, \ldots.

For irregular primes a more delicate argument is needed. Vandiver gave several supplementary criteria; the relevant one here is Vandiver's Theorem III: let p be an irregular prime with p \equiv 1 \pmod 4. If for every irregular Bernoulli index 2k of p (i.e. every k in [1, (p-3)/2] with p \mid \mathrm{num}(B_{2k})) the integer k itself is even, then \mathrm{FLT}_p holds.

For p = 37 the unique irregular index is 2k = 32, so k = 16, which is even, and 37 \equiv 1 \pmod 4. So Vandiver's Theorem III applies at 37, and proving \mathrm{FLT}_{37} reduces to formalising Vandiver III at \ell = 37.