Regular prime

Remarks

Kummer'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 $B_k$ for $k = 2, 4, \ldots, p - 3$. Kummer 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. It is known that there are infinitely many irregular primes. Whether there are infinitely many regular primes is an open question.

Example: $p = 5$ is regular. By 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 $1$.

Non-example: $p = 37$ is irregular. The numerator of $B_{32}$ is divisible by $37$, and $32 \le 37 - 3$; equivalently, $37$ divides the class number of $\Q(\zeta_{37})$.