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

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).

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.

((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 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]

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(ζ_p), where ζ_p 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(ζ_p) up to equivalence. Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζ_p) so that I = uJ. The first few of these class numbers are listed in OEIS: A000927.

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 B_k 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 the numerator of one of these Bernoulli numbers.

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.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 probability

\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}\approx 0.606531

.

It follows that about $60.6531\%$ of the primes are regular by chance. Hart et al.^[4] indicate that $60.6590\%$ of the primes less than $2^{31}=2,147,483,648$ are regular.

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, ... (sequence A000928 in the OEIS)

Infinitude

K. L. Jensen (a student of Nielsen^[5]) proved in 1915 that there are infinitely many irregular primes of the form 4n + 3.^[6] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.^[7]

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,^[8] and later generalized this.^[9]

Irregular pairs

If p is an irregular prime and p divides the numerator of the Bernoulli number B_2k 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), ... (sequence A189683 in the OEIS).

The smallest even k such that nth irregular prime divides B_k 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.^[10] 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 B_2k 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 (or E-irregular) as a prime p that divides at least one Euler number E_2n 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 in 1940 that Fermat's Last Theorem (x^p + y^p = z^p) has no solution for integers x, y, z with gcd(xyz, p) = 1 if p is Euler-regular. Gut proved that x^2p + y^2p = z^2p has no solution if p has an E-irregularity index less than 5.^[11]

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, ... (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 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, ... (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 4pth cyclotomic field Q(ζ_4p).

Weak irregular pairs

In this section, "a_n" means the numerator of the nth Bernoulli number if n is even, "a_n" means the (n − 1)th Euler number if n is odd (sequence A246006 in the OEIS).

Since for every odd prime p, p divides a_p 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 a_p−1. Besides, if and only if an odd prime p divides a_n (and 2p does not divide n), then p also divides a_n+k(p−1) (if 2p divides n, then the sentence should be changed to "p also divides a_n+2kp". In fact, if 2p divides n and p(p − 1) does not divide n, then p divides a_n.) 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 a_18k+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	integers 0 ≤ n ≤ p − 2 such that p divides a_n	p	integers 0 ≤ n ≤ p − 2 such that p divides a_n	p	integers 0 ≤ n ≤ p − 2 such that p divides a_n	p	integers 0 ≤ n ≤ p − 2 such that p divides a_n	p	integers 0 ≤ n ≤ p − 2 such that p divides a_n	p	integers 0 ≤ n ≤ p − 2 such that p divides a_n
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 ≤ n ≤ p − 2 such that p divides a_n.)

The following table shows all irregular pairs with n ≤ 63. (To get these irregular pairs, we only need to factorize a_n. 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 ^[12]).

n	primes p ≥ n + 2 such that p divides a_n	n	primes p ≥ n + 2 such that p divides a_n
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 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.

n	primes p such that p divides a_p−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, ...

Wolstenholme prime

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[Gardiner1988-1] [1]
Gardiner, A. (1988), "Four Problems on Prime Power Divisibility", American Mathematical Monthly, 95 (10): 926–931, doi:10.2307/2322386, JSTOR 2322386

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