# Regular prime

 Are there infinitely many regular primes, and if so is their relative density $e^{-1/2}$?

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. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

The first few regular odd primes are:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, ... (sequence A007703 in OEIS).

## Definition

### Class number criterion

An odd prime number p is defined to be regular if it does not divide the class number of the p-th cyclotomic field Qp), where ζp is a p-th root of unity. The prime number 2 is often considered regular as well.

The class number of the cyclotomic field is the number of ideals of the ring of integers Zp) up to isomorphism. Two ideals I,J are considered isomorphic if there is a nonzero u in Qp) so that I=uJ.

### Kummer's criterion

Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bk for k = 2, 4, 6, …, p − 3.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of p dividing one of these Bernoulli numbers.

## Siegel's conjecture

It has been conjectured that there are infinitely many regular primes. 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. Neither conjecture has been proven since their conception.

## Irregular primes

An odd prime that is not regular is an irregular prime. The first few irregular primes are:

37, 59, 67, 101, 103, 131, 149, ... (sequence A000928 in OEIS)

### Infinitude

K. L. Jensen (an unknown student of Nielsen[1]) has shown in 1915 that there are infinitely many irregular primes of the form 4n + 3. [2] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.[3]

Metsänkylä proved[4] that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1.

### Irregular pairs

If p is an irregular prime and p divides the numerator of the Bernoulli number B2k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. In other words, an irregular pair is a book-keeping device to record, for an irregular prime p, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs are:

(691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), ... (sequence A189683 in OEIS).

For a given prime p, the number of such pairs is called the index of irregularity of p.[5] Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.

It was discovered that (p, p − 3) is in fact an irregular pair for p = 16843. This is the first and only time this occurs for p < 30000.

## History

In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p is regular. This raised attention in the irregular primes.[6] 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. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair.

Kummer found the irregular primes less than 165. 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.[7]