# regular prime

A prime $p$ is 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 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.

## References

• 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, Second Edition, 1990.
Title regular prime RegularPrime 2013-03-22 12:44:20 2013-03-22 12:44:20 djao (24) djao (24) 6 djao (24) Definition msc 11R18 msc 11R29 irregular prime