In mathematics, a regular prime is a certain kind of odd number prime number. A prime number p is 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). The first few regular primes are: :3 (number), 5 (number), 7 (number), 11 (number), 13 (number), 17 (number), 19 (number), 23 (number), 29 (number), 31 (number), 41 (number), …

It has been conjectured that there are infinity many regular primes. More precisely it is expected that e (mathematical constant) −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 2004.

Historically, regular primes were first considered by Ernst 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 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.

An odd prime that is not regular is an irregular prime. The number of B k with a numerator divisible by p is called the irregularity index of p . Johan Jensen has shown in 1915 that there are infinitely many irregular primes, the first few of which are : :37 (number), 59 (number), 67 (number), 101 (number), 103 (number), 131 (number), 149 (number), …

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