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arXiv · 2305.05522

On Power Sums and the Kummer Congruences

Samuel Goodman
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Abstract

In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates an underlying structure on the level of elements that yields a new proof of the classical Kummer Congruences. Hence the Kummer Congruences can be viewed as a consequence of putting together this local information, the interactions between which are understood. This gives a new perspective in that all known approaches are based directly on global considerations, namely either through global sum identities or $p$-adic integrals.

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Samuel Goodman. 2023-05-09. On Power Sums and the Kummer Congruences. https://arxiv.org/abs/2305.05522

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