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A generalization of rummer’s congruences

A generalization of rummer’s congruences Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and $$(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )$$ ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN, $$\frac{1}{m}\{ B_m (\frac{\alpha }{k}) - p^{m - 1} B_m (\frac{\beta }{k})\} \equiv \frac{1}{n}\{ B_n (\frac{\alpha }{k}) - p^{n - 1} B_n (\frac{\beta }{k})\} (\bmod p^{N + 1} )$$ ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (6)

Publisher
Springer Journals
Copyright
Copyright © 1997 by Mathematische Seminar
Subject
Mathematics; Algebra; Differential Geometry; Combinatorics; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02940825
Publisher site
See Article on Publisher Site

Abstract

Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and $$(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )$$ ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN, $$\frac{1}{m}\{ B_m (\frac{\alpha }{k}) - p^{m - 1} B_m (\frac{\beta }{k})\} \equiv \frac{1}{n}\{ B_n (\frac{\alpha }{k}) - p^{n - 1} B_n (\frac{\beta }{k})\} (\bmod p^{N + 1} )$$ ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 27, 2008

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