On a special congruence of Carlitz

Sandro Mattarei

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On a special congruence of Carlitz

Sandro Mattarei

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Abstract

We prove that if $q$ is a power of a prime $p$ and $p^k$ divides $a$, with $k\ge 0$, then \[ 1+(q-1)\sum_{0\le b(q-1)<a} \binom{a}{b(q-1)}\equiv 0\pmod{p^{k+1}}. \] The special case of this congruence where $q=p$ was proved by Carlitz in 1953 by means of rather deep properties of the Bernoulli numbers. A more direct approach produces our generalization and several related results.

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Sandro Mattarei. 2006-02-01. On a special congruence of Carlitz. https://arxiv.org/abs/math/0602012

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