# 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 numbersHTML articles powered by AMS MathViewer

by Bernd C. Kellner
Math. Comp. 76 (2007), 405-441 Request permission

## Abstract:

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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