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:
: 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, ... .

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 (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair.

( is an irregular pair when ''p'' is irregular due to a certain condition, described below, being realized at 2''k''.)

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(''ζ''_''p''), where ''ζ''_''p'' is a primitive ''p''th 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 . The first few of these class numbers are listed in .

Kummer's criterion

Ernst Kummer showed that an equivalent criterion for regularity is that ''p'' does not divide the numerator of any of the Bernoulli numbers ''B''_''k'' 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 the numerator of 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.

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

:$\dfrac$

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime $p$ is

:$\left(\dfrac\right)^=\left(1-\dfrac\right)^=\left(1-\dfrac\right)^\cdot\left\lbrace\left(1-\dfrac\right)^\right\rbrace^$.

By the definition of ''e'', we have

:$\lim_\left(1-\dfrac\right)^=\dfrac$

so that we obtain the probability

:$\lim_\left(1-\dfrac\right)^\cdot\left\lbrace\left(1-\dfrac\right)^\right\rbrace^=e^\approx0.606531$.

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

Irregular primes

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, ...

Infinitude

K. L. Jensen (a student of Niels Nielsen) proved in 1915 that there are infinitely many irregular primes of the form .
In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.

Metsänkylä proved in 1971 that for any integer , there are infinitely many irregular primes not of the form or , and later generalized this.

Irregular pairs

If ''p'' is an irregular prime and ''p'' divides the numerator of the Bernoulli number ''B''_2''k'' for , then 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), ... .

The smallest even ''k'' such that ''n''th 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, ...

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 ''B''_2''k'' and these ''k''s are less than . The first irregular prime with irregular index greater than 1 is 157, which divides ''B''₆₂ and ''B''₁₁₀, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

The irregular index of the ''n''th 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)

The irregular index of the ''n''th 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, ...

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, ...

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, ...

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, ...

The least primes having irregular index ''n'' are
: 2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ... (This sequence defines "the irregular index of 2" as −1, and also starts at .)

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 ''E''_2''n'' with . 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, ...

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 () has no solution for integers ''x'', ''y'', ''z'' with if ''p'' is Euler-regular. Gut proved that 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 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, ...

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 , , , , , and are also all composite ( Legendre proved the first case of Fermat's Last Theorem for primes ''p'' such that at least one of , , , , , and 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, ...

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 4''p''th cyclotomic field Q(''ζ''_4''p'').

Weak irregular pairs

In this section, "''a_n''" means the numerator of the ''n''th Bernoulli number if ''n'' is even, "''a_n''" means the th Euler number if ''n'' is odd .

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 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 2''p'' does not divide ''n''), then ''p'' also divides ''a''_{''n''+''k''(''p''−1)} (if 2''p'' divides ''n'', then the sentence should be changed to "''p'' also divides ''a''_{''n''+2''kp''}". In fact, if 2''p'' divides ''n'' and does not divide ''n'', then ''p'' divides ''a''_''n''.) for every integer ''k'' (a condition is must be > 1). For example, since 19 divides ''a''₁₁ and does not divide 11, so 19 divides ''a''_18''k''+11 for all ''k''. Thus, the definition of irregular pair , ''n'' should be at most .

The following table shows all irregular pairs with odd prime :

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 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, , but , so the only irregular pair with is ) (for more information (even ''n''s up to 300 and odd ''n''s up to 201), see ).

The following table shows irregular pairs (), it is a conjecture that there are infinitely many irregular pairs for every natural number , but only few were found for fixed ''n''. For some values of ''n'', even there is no known such prime ''p''.

See also

* Wolstenholme prime

References

Further reading

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External links

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* Chris Caldwell
The Prime Glossary: regular prime
at The Prime Pages.
* Keith Conrad
Fermat's last theorem for regular primes


Bernoulli irregular prime

Euler irregular prime






Factorization of Bernoulli and Euler numbers

{{Prime number classes

Algebraic number theory
Cyclotomic fields
Classes of prime numbers
Unsolved problems in number theory