Regular prime

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In number theory, a regular prime is a prime number p that does not divide the class number of the p-th cyclotomic field, and is called irregular otherwise. Ernst Kummer showed that an equivalent criterion for regularity (for odd primes) 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 irregular primes are: 37, 59, 67, 101, ... (sequence A000928 in OEIS)

The regular primes are (sequence A007703 in 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 2009^[update].

Regular primes were first considered by Kummer, who was able to prove that Fermat's last theorem holds true for regular prime exponents.

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.

[edit] 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.

[edit] See also

• Herbrand–Ribet theorem

[edit] External links

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