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 theDefinition
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, 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 theKummer's criterion
Siegel's conjecture
It has beenIrregular 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, ...Infinitude
K. L. Jensen (an otherwise unknown student of 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 that for any integer , there are infinitely many irregular primes not of the form or , and later generalized it.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 ''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, ... 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''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 . 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 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'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, ... 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'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, ... 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''''4p'').Weak irregular pairs
In this section, "''an''" means the numerator of the ''n''th Bernoulli number if ''n'' is even, "''an''" means the 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 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 ''an'' (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 ''an''.) The following table shows all irregular pairs with ''n'' ≤ 63. (To get these irregular pairs, we only need to factorize ''an''. 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 primeReferences
Further reading
* * * * * * * * * * * * * *External links
* * Chris Caldwell