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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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On irregular prime power divisors of the Bernoulli numbers
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by Bernd C. Kellner PDF
Math. Comp. 76 (2007), 405-441 Request permission


Let $B_n$ ($n = 0, 1, 2, \ldots$) denote the usual $n$th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not \equiv B_{l+p-1}/(l+p-1) \operatorname {mod}{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \operatorname {mod}{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \operatorname {mod}{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $\chi _{(p, l)}$ which is a zero of a certain $p$-adic zeta function $\zeta _{p, l}$ originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) $p$-adic expansion of $\chi _{(p, l)}$ for irregular pairs $(p,l)$ with $p$ below 1000.
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Additional Information
  • Bernd C. Kellner
  • Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstr. 3–5, 37073 Göttingen, Germany
  • Email:
  • Received by editor(s): June 15, 2005
  • Received by editor(s) in revised form: September 4, 2005
  • Published electronically: August 1, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 405-441
  • MSC (2000): Primary 11B68; Secondary 11M06, 11R23
  • DOI:
  • MathSciNet review: 2261029