regular prime

A prime p is regularPlanetmathPlanetmath if the class numberMathworldPlanetmathPlanetmath of the cyclotomic fieldMathworldPlanetmath (ζp) is not divisible by p (where ζp:=e2πi/p denotes a primitive pth root of unityMathworldPlanetmath). An irregular primeMathworldPlanetmath 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 TheoremMathworldPlanetmath. 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:

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 infiniteMathworldPlanetmath, although it is known that there are infinitely many irregular primes.


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