# A New Generalisation of the Kummer Congruence

• H. Sunil Gunaratne
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 325)

## Abstract

The following generalization
$${G_c}(j,w,n) = - (\begin{array}{*{20}{c}} {{p^{ - 1}}{\Delta _c}} \\ j \end{array})(1 - {w^{m - n}}(p){p^{n - 1}})\frac{1}{n}{B_{n,{w^{m - n}}}} \in {Z_p}$$
of the Kummer congruence is obtained. Here Δ c is the difference operator Δ c x n = x n+c x n , B n, χ is the generalised Bernoulli number of Leopoldt and ($$\begin{array}{*{20}{c}} {{p^{ - 1}}{\Delta _c}} \\ j \end{array}$$) is the binomial coefficient operator. This congruence is best possible in the sense that there does not exist a pair (p, m), where ω m (≠1) is even, such that G c (j, ω m , n) ∈ p Z p for all j ≥ 0.
The classical generalisation of the Kummer congruence is
$${K_c}(j,{w^m},n) = {p^{ - j}}\Delta _c^j(1 - {w^{m - n}}(p){p^{n - 1}})\frac{1}{n}{B_{n,{w^{m - n}}}} \in {Z_p}.$$
It will be shown that this is periodic in the sense that K c (j, ω m , n) ≡ K c (j′, ω m , n′) mod p Z p if jj′ mod (p − 1) with j, j′ > 0 and nn′ mod (p − 1).

## Keywords

Power Series Expansion Bernoulli Polynomial Cyclotomic Field Classical Generalisation Elementary Number Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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