regular prime
A prime is regular if the class number
of the cyclotomic field
is not divisible by (where denotes a primitive 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:
- •
is regular if and only if none of the numerators of the Bernoulli numbers
, , is a multiple
of .
Based on this criterion it is possible to give a heuristic argument that the regular primes have density 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, AClassical Introduction to Modern Number Theory
, Springer-Verlag, NewYork, Second Edition, 1990.