Kummer's congruence

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 B_h 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(p^a+1) is the Euler totient function, evaluated at p^a+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 754003
• 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
• Kummer, Ernst Eduard (1851), "Über eine allgemeine Eigenschaft der rationalen Entwicklungscoëfficienten einer bestimmten Gattung analytischer Functionen", Journal für Reine und Angewandte Mathematik 41: 368–372, doi:10.1515/crll.1851.41.368, ISSN 0075-4102, JFM 041.1136cj