Regular prime

In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

The first few regular odd primes are:

Definition

Class number criterion

An odd prime number p is defined to be regular if it does not divide the class number of the p-th cyclotomic field Q(ζ_p), where ζ_p is a p-th root of unity, it is listed on  A000927. The prime number 2 is often considered regular as well.

The class number of the cyclotomic field is the number of ideals of the ring of integers Z(ζ_p) up to isomorphism. Two ideals I,J are considered isomorphic if there is a nonzero u in Q(ζ_p) so that I=uJ.

Kummer's criterion

Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers B_k for k = 2, 4, 6, …, p − 3.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of p dividing one of these Bernoulli numbers.