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Concept
Regular prime
Summary
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, ... .
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 (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. It was found in 1993 that the next time this happens is for p = 2124679; see Wolstenholme prime.
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 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(ζp) up to equivalence. Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζp) so that I = uJ. The first few of these class numbers are listed in .
Official source
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