Proofs of power sum and binomial coefficient congruences via Pascal’s identity. (English) Zbl 1230.05014

MacMillan, Kieren; Sondow, Jonathan

Am. Math. Mon. 118, No. 6, 549-551 (2011).

Summary: A well-known and frequently cited congruence for power sums is \[ 1^n+ 2^n+\cdots+ p^n\equiv\begin{cases} -1\pmod p\;&\text{if }(p-1)\mid n,\\ 0\pmod p\;&\text{if }(p-1)\nmid n,\end{cases} \] 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 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:

11A07	Congruences; primitive roots; residue systems
11B65	Binomial coefficients; factorials; \(q\)-identities
05A10	Factorials, binomial coefficients, combinatorial functions
05A19	Combinatorial identities, bijective combinatorics

Keywords:

congruence for power sums; congruence for sums of binomial coefficients

Citations:

Zbl 0253.10001; JFM 41.0221.10; JFM 07.0131.01