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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 ++p n -1(modp)if(p-1)n,0(modp)if(p-1)n,

where n1 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 B. Pascal in the year 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to Ch. Hermite [J. Reine Angew. Math. 81, 93–95 (1875; JFM 07.0131.01)] and P. Bachmann [Niedere Zahlentheorie. Zweiter Teil, Teubner, Leipzig (1910; JFM 41.0221.10) (p. 53); Reprint. Bronx, N. Y.: Chelsea (1968; Zbl 0253.10001)].

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