# Regular prime

 Are there infinitely many regular primes, and if so is their relative density ${\displaystyle e^{-1/2}}$?

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, ... (sequence A007703 in OEIS).

## 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 p-th root of unity, it is listed on Template:Oeis. 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 Template:Harv 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 Template:Harvs 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 since their conception.

## 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 OEIS)

### Infinitude

K. L. Jensen (an unknown student of Nielsen[1]) has shown in 1915 that there are infinitely many irregular primes of the form 4n + 3. [2] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.[3]

Metsänkylä proved[4] 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 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 OEIS)

For a given prime p, the number of such pairs is called the index of irregularity of p.[5] 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. This is the first and only time this occurs for p < 30000.

### 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

-1, 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, 0, 0, 2, ... (This sequence defines "the irregular index of 2" as -1) (sequence A091888 in 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, 3, 1, 1, ... (sequence A091887 in 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 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 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 OEIS)

The least primes having irregular index n are

2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ...(sequence A061576 in 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 ≤ 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 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.

### Strong Irregular primes

A prime p is strong irregular if and only if p divides the numerator of B2n for some n < ${\displaystyle {\frac {p-1}{2}}}$, and p also divides E2n for some n < ${\displaystyle {\frac {p-1}{2}}}$, (the two ns can be either the same or different), where Bn is the Bernoulli number and En is the Euler number, 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 OEIS)

Clearly, 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 case is that p = 263.

### Weak irregular primes

A prime p is weak irregular if and only if p divides the numerator of B2n or E2n for some n < ${\displaystyle {\frac {p-1}{2}}}$, where Bn is the Bernoulli number and En is the Euler number, 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, ...

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, ... (a2n = the absolute value of the numerator of B2n, and a2n+1 = the absolute value of E2n, this sequence starts at n = 0)

The weak irregular pairs are (For the pairs (p, n) which p divides an, and np-2)

(61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ... (The even ns means the numerator of Bn, and the odd ns means En-1)

The following table shows all weak irregular primes below 300 (Weak irregular index is defined as "Bernoulli irregular index + Euler irregular index")

 p (Weak irregular prime) 19 31 37 43 47 59 61 67 71 79 101 103 131 137 n which p divides an 11 23 32 13 15 44 7 27, 58 29 19 63, 68 24 22 43 Weak irregular index 1 1 1 1 1 1 1 2 1 1 2 1 1 1 p (Weak irregular prime) 139 149 157 193 223 233 241 251 257 263 271 277 283 293 n which p divides an 129 130, 147 52, 110 75 133 84 211, 239 127 164 100, 213 84 9 20 156 Weak irregular index 1 2 2 1 1 1 2 1 1 2 1 1 1 1

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.

### 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 OEIS)

The density of them 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 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 is 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 is 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.[6] 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.[7]

## References

1. Leo Corry: Number Crunching vs. Number Theory: Computers and FLT, from Kummer to SWAC (1850-1960), and beyond
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