A conjecture about numerators of Bernoulli numbers related to Integer Sequence A092291

In this paper we disprove a conjecture about numerators of divided Bernoulli numbers $B_n/n$ and $B_n/n(n-1)$ which was suggested by Roland Bacher. We give some counterexamples. Finally, we extend the results to the general case.Comment: 11 page

On irregular prime power divisors of the Bernoulli numbers

Let $B_n$ ($n = 0, 1, 2, ...$) 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) \modp{p^2}. We show that for every $r \geq 1$ the congruence B_{m_r}/m_r \equiv 0 \modp{p^r} has a unique solution $m_r$ where m_r \equiv l \modp{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.Comment: 42 pages; final accepted paper, slightly revised and extended, to appear in Math. Com