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Proofs of power sum and binomial coefficient congruences via Pascal’s identity. (English) Zbl 1230.05014
Summary: A well-known and frequently cited congruence for power sums is $$1^n+ 2^n+\cdots+ p^n\equiv\cases -1\pmod p\ &\text{if }(p-1)\mid n,\\ 0\pmod p\ &\text{if }(p-1)\nmid n,\endcases$$ where $n\geq 1$ and $p$ is prime. We survey the main ingredients in several known proofs. Then we give an elementary proof, using an identity for power sums proven by {\it B. Pascal} in the year 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to {\it Ch. Hermite} [J. Reine Angew. Math. 81, 93--95 (1875; JFM 07.0131.01)] and {\it P. Bachmann} [Niedere Zahlentheorie. Zweiter Teil, Teubner, Leipzig (1910; JFM 41.0221.10) (p. 53); Reprint. Bronx, N. Y.: Chelsea (1968; Zbl 0253.10001)].

11A07Congruences; primitive roots; residue systems
11B65Binomial coefficients, etc.
05A10Combinatorial functions
05A19Combinatorial identities, bijective combinatorics
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