regular prime

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 $B_0$, $B_2$, $B_4,\mathrm{\dots },B_{p-3}$ 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.

Title	regular prime
Canonical name	RegularPrime
Date of creation	2013-03-22 12:44:20
Last modified on	2013-03-22 12:44:20
Owner	djao (24)
Last modified by	djao (24)
Numerical id	6
Author	djao (24)
Entry type	Definition
Classification	msc 11R18
Classification	msc 11R29
Defines	irregular prime