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.

 Unsolved problem in mathematics:Are there infinitely many regular primes, and if so, is their relative density ${\displaystyle e^{-1/2}}$?(more unsolved problems in mathematics)

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, ... (sequence A007703 in the OEIS).

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.[1] 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.[2] It was found in 1993 that the next time this happens is for p = 2124679, see Wolstenholme prime.[3]

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 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 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 e1/2, 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 (or Bernoulli irregular or B-irregular to distinguish from other types or 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, ... (sequence A000928 in the OEIS)

Infinitude

K. L. Jensen (an otherwise unknown student of Nielsen[4]) proved in 1915 that there are infinitely many irregular primes of the form 4n + 3. [5] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.[6]

Metsänkylä proved[7] 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), (2294797, 24), (657931, 26), (9349, 28), (362903, 28), ... (sequence A189683 in the OEIS).

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, ... (sequence A035112 in the OEIS)

For a given prime p, the number of such pairs is called the index of irregularity of p.[8] 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, as well as for p = 2124679. 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) (sequence A091888 in the OEIS)

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, ... (sequence A091887 in the OEIS)

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, ... (sequence A073276 in the OEIS)

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, ... (sequence A073277 in the OEIS)

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, ... (sequence A060975 in the OEIS)

The least primes having irregular index n are

2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ... (sequence A061576 in the OEIS) (This sequence defines "the irregular index of 2" as −1, and also starts at n = −1.)

Generalizations

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 < 2np − 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, ... (sequence A120337 in the OEIS)

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 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.[9][10]

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 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, ... (sequence A128197 in the OEIS)

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's 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, ... (sequence A250216 in the OEIS)

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 Q4p).

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 (sequence A246006 in the OEIS).

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:

 p integers0 ≤ n ≤ p - 2such that p divides an p integers0 ≤ n ≤ p - 2such that p divides an p integers0 ≤ n ≤ p - 2such that p divides an p integers0 ≤ n ≤ p - 2such that p divides an p integers0 ≤ n ≤ p - 2such that p divides an p integers0 ≤ n ≤ p - 2such that p divides an 3 79 19 181 293 156 421 240 557 222 5 83 191 307 88, 91, 137 431 563 175, 261 7 89 193 75 311 87, 193, 292 433 215, 366 569 11 97 197 313 439 571 389 13 101 63, 68 199 317 443 577 52, 209, 427 17 103 24 211 331 449 587 45, 90, 92 19 11 107 223 133 337 457 593 22 23 109 227 347 280 461 196, 427 599 29 113 229 349 19, 257 463 130, 229 601 31 23 127 233 84 353 71, 186, 300 467 94, 194 607 592 37 32 131 22 239 359 125 479 613 522 41 137 43 241 211, 239 367 487 617 20, 174, 338 43 13 139 129 251 127 373 163 491 292, 336, 338, 429 619 371, 428, 543 47 15 149 130, 147 257 164 379 100, 174, 317 499 631 80, 226 53 151 263 100, 213 383 503 641 59 44 157 62, 110 269 389 200 509 141 643 61 7 163 271 84 397 521 647 236, 242, 554 67 27, 58 167 277 9 401 382 523 400 653 48 71 29 173 281 409 126 541 86, 465 659 224 73 179 283 20 419 159 547 270, 486 661

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 ≤ np - 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 [11])

 n primes p ≥ n + 2 such that p divides an n primes p ≥ n + 2 such that p divides an 0 32 37, 683, 305065927 1 33 930157, 42737921, 52536026741617 2 34 151628697551 3 35 4153, 8429689, 2305820097576334676593 4 36 26315271553053477373 5 37 9257, 73026287, 25355088490684770871 6 38 154210205991661 7 61 39 23489580527043108252017828576198947741 8 40 137616929, 1897170067619 9 277 41 763601, 52778129, 359513962188687126618793 10 42 1520097643918070802691 11 19, 2659 43 137, 5563, 13599529127564174819549339030619651971 12 691 44 59, 8089, 2947939, 1798482437 13 43, 967 45 587, 32027, 9728167327, 36408069989737, 238716161191111 14 46 383799511, 67568238839737 15 47, 4241723 47 285528427091, 1229030085617829967076190070873124909 16 3617 48 653, 56039, 153289748932447906241 17 228135437 49 5516994249383296071214195242422482492286460673697 18 43867 50 417202699, 47464429777438199 19 79, 349, 87224971 51 5639, 1508047, 10546435076057211497, 67494515552598479622918721 20 283, 617 52 577, 58741, 401029177, 4534045619429 21 41737, 354957173 53 1601, 2144617, 537569557577904730817, 429083282746263743638619 22 131, 593 54 39409, 660183281, 1120412849144121779 23 31, 1567103, 1427513357 55 2749, 3886651, 78383747632327, 209560784826737564385795230911608079 24 103, 2294797 56 113161, 163979, 19088082706840550550313 25 2137, 111691689741601 57 5303, 7256152441, 52327916441, 2551319957161, 12646529075062293075738167 26 657931 58 67, 186707, 6235242049, 37349583369104129 27 67, 61001082228255580483 59 1459879476771247347961031445001033, 8645932388694028255845384768828577 28 9349, 362903 60 2003, 5549927, 109317926249509865753025015237911 29 71, 30211, 2717447, 77980901 61 6821509, 14922423647156041, 190924415797997235233811858285255904935247 30 1721, 1001259881 62 157, 266689, 329447317, 28765594733083851481 31 15669721, 28178159218598921101 63 101, 6863, 418739, 1042901, 91696392173931715546458327937225591842756597414460291393

The following table shows irregular pairs (p, p - n) (n ≥ 2), it's 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.

 n primes p such that p divides ap - n (these p are checked up to 20000) OEIS sequence 2 149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319, ... A198245 3 16843, 2124679, ... A088164 4 ... 5 37, ... 6 ... 7 ... 8 19, 31, 3701, ... 9 67, 877, ... A212557 10 139, ... 11 9311, ... 12 ... 13 ... 14 ... 15 59, 607, ... 16 1427, 6473, ... 17 2591, ... 18 ... 19 149, 311, 401, 10133, ... 20 9643, ... 21 8369, ... 22 ... 23 ... 24 17011, ... 25 ... 26 ... 27 ... 28 ... 29 4219, 9133, ... 30 43, 241, ... 31 3323, ... 32 47, ... 33 101, 2267, ... 34 461, ... 35 ... 36 1663, ... 37 ... 38 101, 5147, ... 39 3181, 3529, ... 40 67, 751, 16007, ... 41 773, ...

Harmonic irregular primes

A prime p such that p divides Hk for some 1≤kp-2 is called Harmonic irregular primes (since p (In fact, p2) always divides Hp - 1), where Hk is the numerator of the Harmonic numbers , the first of them are

11, 29, 37, 43, 53, 61, 97, 109, 137, 173, 199, 227, 257, 269, 271, 313, 347, 353, 379, 397, 401, 409, 421, 433, 439, 509, 521, 577, 599, ... (sequence A092194 in the OEIS)

The density of them (to the set of primes) is about 0.367879..., very close to that of B-irregular or E-irregular primes.

The numerator of the Harmonic numbers (also called Wolstenholme numbers) are

0, 1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 55835135, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387, ... (this sequence starts at n = 0) (sequence A001008 in the OEIS)

The Harmonic irregular pairs are

(11, 3), (137, 5), (11, 7), (761, 8), (7129, 9), (61, 10), (97, 11), (863, 11), (509, 12), (29, 13), (43, 13), (919, 13), (1049, 14), (1117, 14), (29, 15), (41233, 15), (8431, 16), (37, 17), (1138979, 17), (39541, 18), (37, 19), (7440427, 19), ...

In fact, if and only if a prime p divides Hk, then p also divides Hp - 1 - k, so all odd prime p have an even Harmonic irregular index (0 is also an even number).

The super irregular primes (odd primes which are B-irregular, E-irregular, and H-irregular) are

353, 379, 433, 577, 677, 761, 773, 821, 929, 971, ...

The super regular primes (odd primes which are B-regular, E-regular, and H-regular) are

3, 5, 7, 13, 17, 23, 41, 73, 83, 89, 107, 113, 127, 151, 163, 167, 179, 181, 191, 197, 211, 229, 239, 281, 317, 331, 337, 367, 383, 431, 443, 449, 457, 479, 487, 503, 569, ...