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Wikipedia / Regular prime

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:

History and motivation

In 1850, Kummer proved that Fermat's Last Theorem is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem 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 fails to be an irregular pair. (As applied in these results, .)

.; see Wolstenholme prime.

Definition

Class number criterion

is defined to be regular if it does not divide the class numberth cyclotomic fieldth 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. The first few of these class numbers are listed in .

Kummer's criterion

Ernst Kummer does not divide the numerator of any of the Bernoulli numbers.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem dividing the numerator of one of these Bernoulli numbers.

Siegel's conjecture

It has been conjectured that there are infinitely many regular primes. More precisely Carl Ludwig Siegel, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density is the base of the natural logarithm.

is

is

of the primes are regular by chance. Hart et al. are regular.

Irregular primes

An odd prime that is not regular is an irregular prime (or Bernoulli irregular or B-irregular to distinguish from other types of irregularity discussed below). The first few irregular primes are:

  1. 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, ...

Infinitude

K. L. Jensen (a student of Niels Nielsen. In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.

, and later generalized this.

Irregular pairs

is called an irregular pair) are:

are

, the number of such pairs is called the index of irregularity. 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.

.

Irregular index

has irregular index if and only if. The first irregular prime with irregular index greater than 1 is 157, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

, or the prime 3 is

th irregular prime is

The primes having irregular index 1 are

The primes having irregular index 2 are

The primes having irregular index 3 are

are .)

Generalizations

Euler irregular primes

Similarly, we can define an Euler irregular prime that divides at least one Euler number. The first few Euler irregular primes are

The Euler irregular pairs are

Vandiver proved in 1940 that Fermat's Last Theorem has an E-irregularity index less than 5.

See also

References

Further reading