- Regular prime
In

number theory , a**regular prime**is a certain kind ofprime number . A prime number "p" is called regular if it does not divide the class number of the "p"-thcyclotomic field (that is, thealgebraic number field obtained by adjoining the "p"-throot of unity to therational number s).Ernst Kummer showed that an equivalent criterion for regularity is that "p" does not divide thenumerator of any of theBernoulli number s "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

conjecture d 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 theasymptotic sense ofnatural density . Neither conjecture has been provenas 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.**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.**ee also***

Herbrand–Ribet theorem **External links*** Chris Caldwell, [

*http://primes.utm.edu/glossary/page.php?sort=Regular The Prime Glossary: regular prime*] at ThePrime Pages .

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