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:
 * 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ....

History and motivation
In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p 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 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.

((p, 2k) is an irregular pair when p is irregular due to a certain condition described below being realized at 2k.)

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 and that this is the first and only time this occurs for p < 30000. It was found in 1993 that the next time this happens is for ; see Wolstenholme prime.

Class number criterion
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.

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 Z(ζp) up to equivalence. Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζp) so that I = uJ. The first few of these class numbers are listed in.

Kummer's criterion
Ernst Kummer showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bk for.

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 the numerator of one of these Bernoulli numbers.

Siegel's conjecture
It has been conjectured that there are infinitely many regular primes. More precisely 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 $$B_k$$, $$k=2,\dots,p-3$$, is not divisible by the prime $$p$$ is


 * $$\dfrac{p-1}{p}$$

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime $$p$$ is


 * $$\left(\dfrac{p-1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}$$.

By E_(mathematical_constant), we have


 * $$\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{p}=\dfrac{1}{e}$$

so that we obtain the probabilty


 * $$\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}=e^{-1/2}\approx0.606531$$.

It follows that about $$60.6531\%$$ of the primes are regular by chance. Hart et al. indicate that $$60.6590\%$$ of the primes less than $$2^{31}=2,147,483,648$$ 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:
 * 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 Nielsen ) proved 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.

Metsänkylä proved in 1971 that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1, and later generalized this.

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 bookkeeping device to record, for an irregular prime p, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs (when ordered by k) are:
 * (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), (2294797, 24), (657931, 26), (9349, 28), (362903, 28), ....

The smallest even k such that nth irregular prime divides ''Bk are
 * 32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, ...

For a given prime p, the number of such pairs is called the index of irregularity of p. 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, as well as for. There are no more occurrences for p < 109.

Irregular index
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 irregular prime with irregular index greater than 1 is 157, which divides B62 and B110, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

The irregular index of the nth prime is
 * 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, ... (Start with n = 2, or the prime = 3)

The irregular index of the nth irregular prime is
 * 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, ...

The primes having irregular index 1 are
 * 37, 59, 67, 101, 103, 131, 149, 233, 257, 263, 271, 283, 293, 307, 311, 347, 389, 401, 409, 421, 433, 461, 463, 523, 541, 557, 577, 593, 607, 613, 619, 653, 659, 677, 683, 727, 751, 757, 761, 773, 797, 811, 821, 827, 839, 877, 881, 887, 953, 971, ...

The primes having irregular index 2 are
 * 157, 353, 379, 467, 547, 587, 631, 673, 691, 809, 929, 1291, 1297, 1307, 1663, 1669, 1733, 1789, 1933, 1997, 2003, 2087, 2273, 2309, 2371, 2383, 2423, 2441, 2591, 2671, 2789, 2909, 2957, ...

The primes having irregular index 3 are
 * 491, 617, 647, 1151, 1217, 1811, 1847, 2939, 3833, 4003, 4657, 4951, 6763, 7687, 8831, 9011, 10463, 10589, 12073, 13217, 14533, 14737, 14957, 15287, 15787, 15823, 16007, 17681, 17863, 18713, 18869, ...

The least primes having irregular index n are
 * 2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ... (This sequence defines "the irregular index of 2" as −1, and also starts at n = −1.)

Euler irregular primes
Similarly, we can define an Euler irregular prime (or E-irregular) as a prime p that divides at least one Euler number E2n with 0 < 2n ≤ p − 3. The first few Euler irregular primes are
 * 19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, ...

The Euler irregular pairs are
 * (61, 6), (277, 8), (19, 10), (2659, 10), (43, 12), (967, 12), (47, 14), (4241723, 14), (228135437, 16), (79, 18), (349, 18), (84224971, 18), (41737, 20), (354957173, 20), (31, 22), (1567103, 22), (1427513357, 22), (2137, 24), (111691689741601, 24), (67, 26), (61001082228255580483, 26), (71, 28), (30211, 28), (2717447, 28), (77980901, 28), ...

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. Gut proved that x2p + y2p = z2p has no solution if p has an E-irregularity index less than 5.

It was proven that there is an infinity of E-irregular primes. A stronger result was obtained: there is an infinity of E-irregular primes congruent to 1 modulo 8. As in the case of Kummer's B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.

Strong irregular primes
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). The first few strong irregular primes are
 * 67, 101, 149, 263, 307, 311, 353, 379, 433, 461, 463, 491, 541, 577, 587, 619, 677, 691, 751, 761, 773, 811, 821, 877, 887, 929, 971, 1151, 1229, 1279, 1283, 1291, 1307, 1319, 1381, 1409, 1429, 1439, ...

To prove the Fermat's Last Theorem for a strong irregular prime p is more difficult (since Kummer proved the first case of Fermat's Last Theorem for B-regular primes, Vandiver proved the first case of Fermat's Last Theorem for E-regular primes), the most difficult is that p is not only a strong irregular prime, but 2p + 1, 4p + 1, 8p + 1, 10p + 1, 14p + 1, and 16p + 1 are also all composite (Legendre proved the first case of Fermat's Last Theorem for primes p such that at least one of 2p + 1, 4p + 1, 8p + 1, 10p + 1, 14p + 1, and 16p + 1 is prime), the first few such p are
 * 263, 311, 379, 461, 463, 541, 751, 773, 887, 971, 1283, ...

Weak irregular primes
A prime p is weak irregular if it is either B-irregular or E-irregular (or both). The first few weak irregular primes are
 * 19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 491, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593, ...

Like the Bernoulli irregularity, the weak regularity relates to the divisibility of class numbers of cyclotomic fields. 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 pairs
In this section, "an" means the numerator of the nth Bernoulli number if n is even, "an" means the (n − 1)th Euler number if n is odd.

Since for every odd prime p, p divides ap if and only if p is congruent to 1 mod 4, and since p divides the denominator of (p − 1)th Bernoulli number for every odd prime p, so for any odd prime p, p cannot divide ap−1. Besides, if and only if an odd prime p divides an (and 2p does not divide n), then p also divides an+k(p−1) (if 2p divides n, then the sentence should be changed to "p also divides an+2kp". In fact, if 2p divides n and p(p − 1) does not divide n, then p divides an.) for every integer k (a condition is n + k(p − 1) must be > 1). For example, since 19 divides a11 and 2 × 19 = 38 does not divide 11, so 19 divides a18k+11 for all k. Thus, the definition of irregular pair (p, n), n should be at most p − 2.

The following table shows all irregular pairs with odd prime p ≤ 661:

The only primes below 1000 with weak irregular index 3 are 307, 311, 353, 379, 577, 587, 617, 619, 647, 691, 751, and 929. Besides, 491 is the only prime below 1000 with weak irregular index 4, and all other odd primes below 1000 with weak irregular index 0, 1, or 2. (Weak irregular index is defined as "number of integers 0 ≤ n ≤ p − 2 such that p divides an.)

The following table shows all irregular pairs with n ≤ 63. (To get these irregular pairs, we only need to factorize an. For example, a34 = 17 × 151628697551, but 17 < 34 + 2, so the only irregular pair with n = 34 is (151628697551, 34)) (for more information (even ns up to 300 and odd ns up to 201), see ).

The following table shows irregular pairs (p, p − n) (n ≥ 2), it is a conjecture that there are infinitely many irregular pairs (p, p − n) for every natural number n ≥ 2, but only few were found for fixed n. For some values of n, even there is no known such prime p.