Regular prime

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, … OEIS|id=A007703.

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, … OEIS|id=A000928.


* 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.

ee also

*Herbrand–Ribet theorem

External links

* Chris Caldwell, [ The Prime Glossary: regular prime] at The Prime Pages.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • regular prime — noun A prime number belonging to one of the two groups that prime numbers are divided into. The other group is the irregular primes …   Wiktionary

  • Regular — The term regular can mean normal or obeying rules. Regular may refer to:In organizations: * Regular Army for usage in the U.S. Army * Regular clergy, members of a religious order subject to a rule of life * Regular Force for usage in the Canadian …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • Regular temperament — is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. The classic example of a regular temperament is meantone temperament,… …   Wikipedia

  • prime minister — prime ministerial /pruym min euh stear ee euhl/, adj. prime ministership, n. prime ministry. the principal minister and head of government in parliamentary systems; chief of the cabinet or ministry: the British prime minister. [1640 50] * * * or… …   Universalium

  • Prime Minister's Questions — (PMQs) (officially Questions to the Prime Minister) is a constitutional convention in the United Kingdom, where every Wednesday when the House of Commons is sitting the Prime Minister spends half an hour answering questions from Members of… …   Wikipedia

  • Prime rate — Prime rate, or Prime Lending Rate, is a term applied in many countries to a reference interest rate used by banks. The term originally indicated the rate of interest at which banks lent to favored customers, i.e., those with high credibility,… …   Wikipedia

  • Prime (liturgy) — Prime, or the First Hour, is a fixed time of prayer of the traditional Divine Office (Canonical Hours), said at the first hour of daylight (approximately 6:00 a.m.), between the morning Hour of Lauds and the 9 a.m. Hour of Terce. It is part of… …   Wikipedia

  • Prime — • Article on the office of Prime, now suppressed Catholic Encyclopedia. Kevin Knight. 2006. Prime     Prime     † …   Catholic encyclopedia

  • Prime Minister parodies (Private Eye) — Prime Minister parodies are a long running feature of the British satirical magazine Private Eye, which have been included in the majority of issues since the magazine s inception. The parodies consist of one arch satirical personification of the …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.