Regular prime

In number theory, a regular prime is a certain kind of prime number. A prime number p is called regular if it does not divide the class number of the p-th cyclotomic field (that is, the algebraic number field obtained by adjoining the p-th root of unity to the rational numbers). Ernst Kummer showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers B_k for k = 2, 4, 6, …, p − 3. The first few regular primes are:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, … (sequence A007703 in the OEIS).

It has been conjectured that there are infinitely many regular primes. More precisely it is conjectured (Siegel, 1964) that e^−1/2, or about 61%, of all prime numbers are regular, in the asymptotic sense of natural density. Neither conjecture has been proven as of 2008.

Historically, regular primes were first considered by Kummer, who was able to prove that Fermat's last theorem holds true for regular prime exponents (and consequently for all exponents that were multiples of regular primes).

An odd prime that is not regular is an irregular prime. The number of Bernoulli numbers B_k with a numerator divisible by p is called the irregularity index of p. K L Jensen has shown in 1915 that there are infinitely many irregular primes, the first few of which are:

37, 59, 67, 101, 103, 131, 149, … (sequence A000928 in the OEIS).

References

• Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section D2.
• Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers. Nachr. Akad. d. Wiss. Goettingen, Math. Phys. K1., II, 1964, 51-62.

External links

• Chris Caldwell, The Prime Glossary: regular prime at The Prime Pages.