A New Generalisation of the Kummer Congruence

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


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).


Power Series Expansion Bernoulli Polynomial Cyclotomic Field Classical Generalisation Elementary Number Theory 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • H. Sunil Gunaratne
    • 1
  1. 1.University of Brunei DarussalamBrunei Darussalam

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