Regular prime
In
mathematics,
regular primes are a certain kind of
prime numbers. A regular prime
p is one that does not divide the class number of the
algebraic number field obtained by adjoining the
p-th root of unity to the
rational numbers; it can be shown that an equivalent criterion is that
p does not divide the
numerator of any of the
Bernoulli numbers Bk for
k ∈ {2, 4, 6, ...,
p − 3}. Regular primes were first described by
Ernst Kummer. A prime that is not regular is called an
irregular prime, and the number of '\'B
k
the numerators of which p
divides is called the irregularity index of p''.
The first few regular primes are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, ... (Sloane's A007703); it has been conjectured that there are infinitely many regular primes and that about e-1/2 of all prime numbers are regular, but neither conjecture has been proven so far. However, it has been shown by Jensen in 1915 that there are infinitely many irregular primes, the first few of which are 37, 59, 67, 101, 103, 131, 149, ... (Sloane's A000928).
Historically, regular primes were considered by Kummer since he was able to prove that Fermat's last theorem held true for exponents that were multiples of regular primes.
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