Bounds of divided universal Bernoulli numbers and universal Kummer congruences
Open Access
- 14 August 2007
- journal article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 136 (1) , 61-71
- https://doi.org/10.1090/s0002-9939-07-09025-9
Abstract
Let be a prime. We obtain good bounds for the -adic sizes of the coefficients of the divided universal Bernoulli number <!-- MATH $\tfrac{\hat{B}_n}{n}$ --> when is divisible by . As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo for the divided universal Bernoulli numbers for the case , which is a new result.
Keywords
This publication has 11 references indexed in Scilit:
- Introduction to 𝑝-adic Analytic Number TheoryPublished by American Mathematical Society (AMS) ,2009
- Universal Kummer congruences mod prime powersJournal of Number Theory, 2004
- A CONGRUENCE FOR FACTORIALSBulletin of the London Mathematical Society, 2004
- Universal Higher Order Bernoulli Numbers and Kummer and Related CongruencesJournal of Number Theory, 2000
- NOTES ON GLAISHER'S CONGRUENCESChinese Annals of Mathematics, Series B, 2000
- p-AdicL-Functions and Sums of PowersJournal of Number Theory, 1998
- The Universal Von Staudt TheoremsTransactions of the American Mathematical Society, 1989
- An analogue of the Von Staudt-Clausen theoremJournal of Algebra, 1984
- The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbersMathematische Annalen, 1975
- Ueber die Entwickelungscoefficienten der lemniscatischen FunctionenMathematische Annalen, 1898