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 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 is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem it is sufficient to establish that either or 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 is in fact an irregular pair for and that this is the first and only time this occurs for. It was found in 1993 that the next time this happens is for ; see Wolstenholme prime.

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 Q, where ζp is a primitive p-th root of unity, it is listed on. 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 up to equivalence. Two ideals I,J are considered equivalent if there is a nonzero u in Q so that I=uJ.

Kummer's criterion

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 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. Neither conjecture has been proven to date.

Irregular primes

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

Infinitude

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 that for any integer T > 6, there are infinitely many irregular primes not of the form or.

Irregular pairs

If p is an irregular prime and p divides the numerator of the Bernoulli number B2k for, then 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:
The smallest even k such that nth irregular prime divides Bk are
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 is in fact an irregular pair for, as well as for. There are no more occurrences for.

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 /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
The irregular index of the nth irregular prime is
The primes having irregular index 1 are
The primes having irregular index 2 are
The primes having irregular index 3 are
The least primes having irregular index n are

Generalizations

Euler irregular primes

Similarly, we can define an Euler irregular prime as a prime p that divides at least one Euler number E2n with 0 < 2n ≤ p − 3. The first few Euler irregular primes are
The Euler irregular pairs are
Vandiver proved that Fermat's Last Theorem has no solution for integers x, y, z with gcd = 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's both B-irregular and E-irregular. The first few strong irregular primes are
To prove the Fermat's Last Theorem for a strong irregular prime p is more difficult, 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, the first few such p are

Weak irregular primes

A prime p is weak irregular if it's either B-irregular or E-irregular. The first few weak irregular primes are
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 4p-th cyclotomic field Q.

Weak irregular pairs

In this section, "an" means the numerator of the nth Bernoulli number if n is even, "an" means the 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 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, then p also divides an + k for every integer k. 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, n should be at most p - 2.
The following table shows all irregular pairs with odd prime p ≤ 661:
pintegers
0 ≤ n ≤ p - 2
such that p divides an		pintegers
0 ≤ n ≤ p - 2
such that p divides an		pintegers
0 ≤ n ≤ p - 2
such that p divides an		pintegers
0 ≤ n ≤ p - 2
such that p divides an		pintegers
0 ≤ n ≤ p - 2
such that p divides an		pintegers
0 ≤ n ≤ p - 2
such that p divides an
37919181293156421240557222
58319130788, 91, 137431563175, 261
7891937531187, 193, 292433215, 366569
1197197313439571389
1310163, 6819931744357752, 209, 427
171032421133144958745, 90, 92
191110722313333745759322
23109227347280461196, 427599
2911322934919, 257463130, 229601
31231272338435371, 186, 30046794, 194607592
373213122239359125479613522
4113743241211, 23936748761720, 174, 338
4313139129251127373163491292, 336, 338, 429619371, 428, 543
4715149130, 147257164379100, 174, 31749963180, 226
53151263100, 213383503641
594415762, 110269389200509141643
61716327184397521647236, 242, 554
6727, 58167277940138252340065348
712917328140912654186, 465659224
7317928320419159547270, 486661

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.
The following table shows all irregular pairs with n ≤ 63:
nprimes p ≥ n + 2 such that p divides annprimes p ≥ n + 2 such that p divides an
03237, 683, 305065927
133930157, 42737921, 52536026741617
234151628697551
3354153, 8429689, 2305820097576334676593
43626315271553053477373
5379257, 73026287, 25355088490684770871
638154210205991661
7613923489580527043108252017828576198947741
840137616929, 1897170067619
927741763601, 52778129, 359513962188687126618793
10421520097643918070802691
1119, 265943137, 5563, 13599529127564174819549339030619651971
126914459, 8089, 2947939, 1798482437
1343, 96745587, 32027, 9728167327, 36408069989737, 238716161191111
1446383799511, 67568238839737
1547, 424172347285528427091, 1229030085617829967076190070873124909
16361748653, 56039, 153289748932447906241
17228135437495516994249383296071214195242422482492286460673697
184386750417202699, 47464429777438199
1979, 349, 87224971515639, 1508047, 10546435076057211497, 67494515552598479622918721
20283, 61752577, 58741, 401029177, 4534045619429
2141737, 354957173531601, 2144617, 537569557577904730817, 429083282746263743638619
22131, 5935439409, 660183281, 1120412849144121779
2331, 1567103, 1427513357552749, 3886651, 78383747632327, 209560784826737564385795230911608079
24103, 229479756113161, 163979, 19088082706840550550313
252137, 111691689741601575303, 7256152441, 52327916441, 2551319957161, 12646529075062293075738167
266579315867, 186707, 6235242049, 37349583369104129
2767, 61001082228255580483591459879476771247347961031445001033, 8645932388694028255845384768828577
289349, 362903602003, 5549927, 109317926249509865753025015237911
2971, 30211, 2717447, 77980901616821509, 14922423647156041, 190924415797997235233811858285255904935247
301721, 100125988162157, 266689, 329447317, 28765594733083851481
3115669721, 2817815921859892110163101, 6863, 418739, 1042901, 91696392173931715546458327937225591842756597414460291393

The following table shows irregular pairs , it's a conjecture that there are infinitely many irregular pairs 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.
nprimes p such that p divides ap - n OEIS sequence
2149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319,...
316843, 2124679,...
4...
537,...
6...
7...
819, 31, 3701,...
967, 877,...
10139,...
119311,...
12...
13...
14...
1559, 607,...
161427, 6473,...
172591,...
18...
19149, 311, 401, 10133,...
209643,...
218369,...
22...
23...
2417011,...
25...
26...
27...
28...
294219, 9133,...
3043, 241,...
313323,...
3247,...
33101, 2267,...
34461,...
35...
361663,...
37...
38101, 5147,...
393181, 3529,...
4067, 751, 16007,...
41773,...