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.
Contents
- Class number criterion
- Kummers criterion
- Siegels conjecture
- Irregular primes
- Infinitude
- Irregular pairs
- Irregular index
- Euler irregular primes
- Strong irregular primes
- Weak irregular primes
- Weak irregular pairs
- Harmonic irregular primes
- History
- References
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).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(ζp), where ζp is a p-th root of unity, it is listed on A000927. 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 isomorphism. Two ideals I,J are considered isomorphic if there is a nonzero u in Q(ζp) 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 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:
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) 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 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. 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.)Euler irregular primes
Similarly, we can define an Euler irregular prime as a prime p that divides at least one 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, ... (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.
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.
Like B-irregularity, E-irregularity relates to the divisibility of class numbers of cyclotomic fields.
Strong irregular primes
A prime p is called strong irregular if p divides the numerator of B2n for some 0 ≤ n <
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 and only if p divides the numerator of B2n or E2n for some 0 ≤ n <
The first values of Bernoulli and Euler numbers are
1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 5, 50521, 691, 2702765, 7, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 854513, 69348874393137901, 236364091, 15514534163557086905, 8553103, 4087072509293123892361, 23749461029, 1252259641403629865468285, ... (sequence A246006 in the OEIS) (a2n = the absolute value of the numerator of B2n, and a2n+1 = the absolute value of E2n, this sequence starts at n = 0)We can regard these numbers as the numerator of the generalized Bernoulli numbers
1, 1, 1, 3, 1, 25, 1, 427, 1, 12465, 5, 555731, 691, 35135945, 7, 2990414715, 3617, 329655706465, 43867, 45692713833379, 174611, 1111113564712575, 854513, 1595024111042171723, 236364091, 387863354088927172625, 8553103, 110350957750914345093747, 23749461029, ...(sequence A193472 in the OEIS) (start with n = 0)and generalized Euler numbers
1, 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, 370371188237525, 4951498053124096, 69348874393137901, 1015423886506852352, 15514534163557086905, 246921480190207983616, 4087072509293123892361, 70251601603943959887872, 1252259641403629865468285, ... (sequence A000111 in the OEIS) (start with n = 0)In fact, if we let Bn be the nth generalized Bernoulli number, En be the nth generalized Euler number, then
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'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.
Harmonic irregular primes
A prime p such that p divides Hk for some 1≤k≤p-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, ...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. 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. It was found in 1993 that the next time this happens is for p = 2124679, see Wolstenholme prime.