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THE UNIVERSAL KUMMER CONGRUENCES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 22 March 2013, pp. 106-132
- Print publication:
- February 2013
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- Article
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- You have access
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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 
$p- 1$.
