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:
History and motivation In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p {\displaystyle p} if p {\displaystyle 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 {\displaystyle p} , if ( p , p − 3 ) {\displaystyle (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 ) {\displaystyle (p,p-3)} or ( p , p − 5 ) {\displaystyle (p,p-5)} fails to be an irregular pair. (As applied in these results, ( p , 2 k ) {\displaystyle (p,2k)} is an irregular pair when p {\displaystyle p} is irregular due to a certain condition, described below, being realized at 2 k {\displaystyle 2k} .) Kummer found the irregular primes smaller 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 ) {\displaystyle (p,p-3)} is in fact an irregular pair for p = 16843 {\displaystyle p=16843} and that this is the first and only time this occurs for p < 30000 {\displaystyle p<30000} . It was found in 1993 that the next time this happens is for p = 2124679 {\displaystyle p=2124679} ; see Wolstenholme prime.
Definition
Class number criterion An odd prime number p {\displaystyle p} is defined to be regular if it does not divide the class number of the p {\displaystyle p} th cyclotomic field Q ( ζ p ) {\displaystyle \mathbb {Q} (\zeta _{p})} , where ζ p {\displaystyle \zeta _{p}} is a primitive p {\displaystyle 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 ) {\displaystyle \mathbb {Z} (\zeta _{p})} up to equivalence. Two ideals I {\displaystyle I} and J {\displaystyle J} are considered equivalent if there is a nonzero u {\displaystyle u} in Q ( ζ p ) {\displaystyle \mathbb {Q} (\zeta _{p})} so that I = u J {\displaystyle I=uJ} . The first few of these class numbers are listed in (sequence A000927 in the OEIS).
Kummer's criterion
Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that p {\displaystyle p} does not divide the numerator of any of the Bernoulli numbers B k {\displaystyle B_{k}} for k = 2 , 4 , 6 , … , p − 3 {\displaystyle k=2,4,6,\dots ,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 {\displaystyle 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 Carl Ludwig Siegel conjectured that e − 1 / 2 {\displaystyle e^{-1/2}} , or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Here, e ≈ 2.718 {\displaystyle e\approx 2.718} is the base of the natural logarithm. Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers B k {\displaystyle B_{k}} , k = 2 , … , p − 3 {\displaystyle k=2,\dots ,p-3} , is not divisible by the prime p {\displaystyle p} is
p − 1 p {\displaystyle {\dfrac {p-1}{p}}}
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