# Regular prime

In number theory, a regular prime is a prime number p > 2 that does not divide the class number of the p-th cyclotomic field.

The first few regular primes are:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, ... (sequence A007703 in OEIS)
 Are there infinitely many regular primes, and if so is their relative density $e^{-1/2}$?

## 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.

## Properties

It has been conjectured that there are infinitely many regular primes. More precisely Siegel conjectured (1964) 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)

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

Metsänkylä proved[2] 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.[3] 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.[4] 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.[5]

## References

1. ^ Carlitz, L. (1954). "Note on irregular primes". Proceedings of the American Mathematical Society (AMS) 5: 329–331. doi:10.1090/S0002-9939-1954-0061124-6. ISSN 1088-6826. MR 61124.
2. ^ Tauno Metsänkylä (1971). "Note on the distribution of irregular primes". Ann. Acad. Sci. Fenn. Ser. A I 492. MR 0274403.
3. ^ Narkiewicz, Władysław (1990), Elementary and analytic theory of algebraic numbers (2nd, substantially revised and extended ed.), Springer-Verlag; PWN-Polish Scientific Publishers, p. 475, ISBN 3-540-51250-0, Zbl 0717.11045
4. ^ Gardiner, A. (1988), "Four Problems on Prime Power Divisibility", American Mathematical Monthly 95 (10): 926–931, doi:10.2307/2322386
5. ^ Johnson, W. (1975), "Irregular Primes and Cyclotomic Invariants", Mathematics of Computation 29 (129): 113–120 Archived at WebCite