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Bounds of divided universal Bernoulli numbers and universal Kummer congruences


Authors: Arnold Adelberg, Shaofang Hong and Wenli Ren
Journal: Proc. Amer. Math. Soc. 136 (2008), 61-71
MSC (2000): Primary 11B68, 11B83; Secondary 11A07
DOI: https://doi.org/10.1090/S0002-9939-07-09025-9
Published electronically: August 14, 2007
MathSciNet review: 2350389
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Abstract: Let $ p$ be a prime. We obtain good bounds for the $ p$-adic sizes of the coefficients of the divided universal Bernoulli number $ \tfrac{\hat{B}_n}{n}$ when $ n$ is divisible by $ p-1$. As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo $ p$ for the divided universal Bernoulli numbers for the case $ (p-1)\vert n$, which is a new result.


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Additional Information

Arnold Adelberg
Affiliation: Department of Mathematics, Grinnell College, Grinnell, Iowa 50112-0806
Email: adelbe@math.grinnell.edu

Shaofang Hong
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: s-f.hong@tom.com, hongsf02@yahoo.com

Wenli Ren
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China, and Department of Mathematics, Dezhou University, Dezhou 253023, People’s Republic of China
Email: renwenli80@163.com

DOI: https://doi.org/10.1090/S0002-9939-07-09025-9
Keywords: Divided universal Bernoulli numbers, universal von Staudt theorem, universal Kummer congruence, $p$-adic valuation
Received by editor(s): July 5, 2006
Received by editor(s) in revised form: December 1, 2006
Published electronically: August 14, 2007
Additional Notes: The second author is the corresponding author and was supported by New Century Excellent Talents in University Grant # NCET-06-0785, and by SRF for ROCS, SEM
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society

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