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
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