ART

In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851).

Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function.
Statement

The simplest form of Kummer's congruence states that

\( {\frac {B_{h}}{h}}\equiv {\frac {B_{k}}{k}}{\pmod p}{\text{ whenever }}h\equiv k{\pmod {p-1}} \)

where p is a prime, h and k are positive even integers not divisible by p−1 and the numbers Bh are Bernoulli numbers.

More generally if h and k are positive even integers not divisible by p − 1, then

\( (1-p^{{h-1}}){\frac {B_{h}}{h}}\equiv (1-p^{{k-1}}){\frac {B_{k}}{k}}{\pmod {p^{{a+1}}}} \)

whenever

\( h\equiv k{\pmod {\varphi (p^{{a+1}})}} \)

where φ(pa+1) is the Euler totient function, evaluated at pa+1 and a is a non negative integer. At a = 0, the expression takes the simpler form, as seen above. The two sides of the Kummer congruence are essentially values of the p-adic zeta function, and the Kummer congruences imply that the p-adic zeta function for negative integers is continuous, so can be extended by continuity to all p-adic integers.
See also

Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers

References
Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3, MR 0754003
Kubota, Tomio; Leopoldt, Heinrich-Wolfgang (1964), "Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen", Journal für die reine und angewandte Mathematik, 214/215: 328–339, doi:10.1515/crll.1964.214-215.328, ISSN 0075-4102, MR 0163900[permanent dead link]
Kummer, Ernst Eduard (1851), "Über eine allgemeine Eigenschaft der rationalen Entwicklungscoëfficienten einer bestimmten Gattung analytischer Functionen", Journal für die Reine und Angewandte Mathematik, 41: 368–372, doi:10.1515/crll.1851.41.368, ISSN 0075-4102, ERAM 041.1136cj

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License