In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem.
[1] In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if (p, p − 3) is not an irregular pair.
In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000.
Although the two latter tables did not appear in print, Johnson found that (p, p − 3) is in fact an irregular pair for p = 16843 and that this is the first and only time this occurs for p < 30000.
[3] An odd prime number p is defined to be regular if it does not divide the class number of the pth cyclotomic field Q(ζp), where ζp is a primitive pth root of unity.
Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζp) so that I = uJ.
More precisely Carl Ludwig Siegel (1964) conjectured that e−1/2, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density.
Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers
is so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime
[6] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.
An odd prime p has irregular index n if and only if there are n values of k for which p divides B2k and these ks are less than (p − 1)/2.
The first few Euler irregular primes are The Euler irregular pairs are Vandiver proved in 1940 that Fermat's Last Theorem (xp + yp = zp) has no solution for integers x, y, z with gcd(xyz, p) = 1 if p is Euler-regular.
A stronger result was obtained: there is an infinity of E-irregular primes congruent to 1 modulo 8.
A prime p is called strong irregular if it is both B-irregular and E-irregular (the indexes of Bernoulli and Euler numbers that are divisible by p can be either the same or different).
In fact, a prime p is weak irregular if and only if p divides the class number of the 4pth cyclotomic field Q(ζ4p).
(Weak irregular index is defined as "number of integers 0 ≤ n ≤ p − 2 such that p divides an.)