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The universal Kummer congruences. (English) Zbl 1355.11017

Summary: Let \(p\) be a prime. In this paper, we present a detailed \(p\)-adic analysis on factorials and double factorials and their congruences. We give good bounds for the \(p\)-adic sizes of the coefficients of the divided universal Bernoulli number \(B_n/n\) when \(n\) is divisible by \(p-1\). Using these, we then establish the universal Kummer congruences modulo powers of a prime \(p\) for the divided universal Bernoulli numbers \(B_n/n\) when \(n\) is divisible by \(p-1\).

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
11D79 Congruences in many variables
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