Regular prime number

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An odd prime number $p$ such that the ideal class number (cf. Class field theory) of the cyclotomic field $\mathbf{Q}(e^{2\pi i/p})$ is not divisible by $p$. All other odd prime numbers are called irregular (see Irregular prime number).


Another, equivalent but more down-to-earth, definition of regular prime number is as follows. A prime number $p$ is called regular if it does not divide any of the numerators of the Bernoulli numbers $B_1,\ldots,B_{(p-3/2)}$, when these numbers (which are rational) are written as irreducible fractions (see [a1]).

Regular prime numbers are important in connection with Fermat's great (or last) theorem. It is known that $x^p+y^p=z^p$ has no positive integer solutions $x,y,z$ if $p$ is a regular prime number (Kummer's theorem). It is not known if there exist infinitely many regular prime numbers. The number of irregular prime numbers is known to be infinite. For more information see Fermat great theorem.


[a1] D. Shanks, "Solved and unsolved problems in number theory" , Chelsea, reprint (1978)
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Regular prime number. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article