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:

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$p$ is regular if and only if none of the numerators of the Bernoulli numbers^{} ${B}_{0}$, ${B}_{2}$, ${B}_{4},\mathrm{\dots},{B}_{p3}$ 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.
References
 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory^{}, SpringerVerlag, New York, Second Edition, 1990.
Title  regular prime 

Canonical name  RegularPrime 
Date of creation  20130322 12:44:20 
Last modified on  20130322 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 