On a congruence modulo $n^3$ involving two consecutive sums of powers

Meštrović, Romeo

Journal of integer sequences, Tome 17 (2014) no. 8.

Résumé

Summary: For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_{k}(n) := 1^{k} + 2^{k} + \dots + n^{k}$, are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo $n^{3}$ involving two consecutive sums $S_{2k}(n)$ and $S_{2k+1}(n)$. This congruence allows us to establish an equivalent formulation of Giuga's conjecture. Moreover, if $k$ is even and $n \ge 5$ is a prime such that $n -1 \? 2k-2$, then this congruence is satisfied modulo $n^{4}$. This suggests a conjecture about when a prime can be a Wolstenholme prime. We also propose several Giuga-Agoh-like conjectures. Further, we establish two congruences modulo $n^{3}$ for two binomial-type sums involving sums of powers $S_{2i}(n)$ with $i = 0, 1, \dots , k$. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums.

Détail

Classification : 05A10, 11A07, 11A51, 11B50, 11B65, 11B68
Keywords: sum of powers, Bernoulli number, giuga's conjecture, carlitz-von staudt result, von staudt-clausen theorem