Kummer's congruence

Short description: Result in number theory showing congruences involving Bernoulli numbers

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

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

Statement

The simplest form of Kummer's congruence states that

[math]\displaystyle{ \frac{B_h}{h}\equiv \frac{B_k}{k} \pmod p \text{ whenever } h\equiv k \pmod {p-1} }[/math]

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

[math]\displaystyle{ (1-p^{h-1})\frac{B_h}{h}\equiv (1-p^{k-1})\frac{B_k}{k} \pmod {p^{a+1}} }[/math]

whenever

[math]\displaystyle{ h\equiv k\pmod {\varphi(p^{a+1})} }[/math]

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 
• 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, https://gdz.sub.uni-goettingen.de/id/PPN243919689_0214_0215?tify={%22pages%22:%5B334%5D,%22view%22:%22info%22} 
• 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, ERAM 041.1136cj, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002147319