The simplest form of Kummer's congruence states that
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
whenever
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.
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
April 13, 2024
kummer, congruence, mathematics, some, congruences, involving, bernoulli, numbers, found, ernst, eduard, kummer, 1851, kubota, leopoldt, 1964, used, define, adic, zeta, function, statement, editthe, simplest, form, states, that, modp, whenever, modp, displayst. In mathematics Kummer s congruences are some congruences involving Bernoulli numbers found by Ernst Eduard Kummer 1851 Kubota amp Leopoldt 1964 used Kummer s congruences to define the p adic zeta function Statement editThe simplest form of Kummer s congruence states that Bhh Bkk modp whenever h k modp 1 displaystyle frac B h h equiv frac B k k pmod p text whenever h equiv k pmod p 1 nbsp 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 ph 1 Bhh 1 pk 1 Bkk modpa 1 displaystyle 1 p h 1 frac B h h equiv 1 p k 1 frac B k k pmod p a 1 nbsp whenever h k modf pa 1 displaystyle h equiv k pmod varphi p a 1 nbsp where f 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 editVon Staudt Clausen theorem another congruence involving Bernoulli numbersReferences editKoblitz 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 Einfuhrung der p adischen Dirichletschen L Funktionen Journal fur 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 Uber eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen Journal fur die Reine und Angewandte Mathematik 41 368 372 doi 10 1515 crll 1851 41 368 ISSN 0075 4102 ERAM 041 1136cj Retrieved from https en wikipedia org w index php title Kummer 27s congruence amp oldid 1074242095, wikipedia, wiki, book, books, library,
