A prime $p$ is regular if the class number of the cyclotomic field $\mathbb{Q}(\zeta_p)$ is not divisible by $p$ (where $\zeta_p := e^{2 \pi i/p}$ denotes a primitive $p^\mathrm{th}$ root of unity). An irregular prime is a prime that is not regular.

Regular primes rose to prominence as a result of Ernst Kummer's work in the 1850's on Fermat's Last Theorem. Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents. In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular:

• $p$ is regular if and only if none of the numerators of the Bernoulli numbers $B_0$ , $B_2$ , $B_4, \ldots, B_{p-3}$ is a multiple of $p$ .

Based on this criterion it is possible to give a heuristic argument that the regular primes have density $e^{-1/2}$ in the set of all primes [1]. Despite this, there is no known proof that the set of regular primes is infinite, although it is known that there are infinitely many irregular primes.

Bibliography

1

Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, Second Edition, 1990.