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:
Based on this criterion it is possible to give a heuristic argument that the regular primes have density in the set of all primes . 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.