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THE UNIVERSAL KUMMER CONGRUENCES

Part of: Sequences and sets Elementary number theory Diophantine equations

Published online by Cambridge University Press:  22 March 2013

SHAOFANG HONG ,
JIANRONG ZHAO  and
WEI ZHAO
SHAOFANG HONG*
Affiliation:
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, PR China Mathematical College, Sichuan University, Chengdu 610064, PR China
JIANRONG ZHAO
Affiliation:
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, PR China email [email protected]
WEI ZHAO
Affiliation:
Science and Technology on Communication Security Laboratory, Chengdu 610041, PR China email [email protected]
*
✉[email protected][email protected][email protected]
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Abstract

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Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis on factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${B}_{n} / n$ when $n$ is divisible by $p- 1$. Using these, we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${B}_{n} / n$ when $n$ is divisible by $p- 1$.

Keywords

universal Bernoulli numberuniversal Kummer congruencedouble factorialp-adic valuationreduced partition

MSC classification

Secondary: 11D79: Congruences in many variables 11B68: Bernoulli and Euler numbers and polynomials 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 1 , February 2013 , pp. 106 - 132
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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