On a Congruence Modulo n Involving Two Consecutive Sums of Powers

For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k+2k+ · · ·+nk, are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo n3 involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n ≥ 5 is a prime such that n− 1 ∤ 2k − 2, then this congruence is satisfied modulo n4. 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 n3 for two binomial-type sums involving sums of powers S2i(n) with i = 0, 1, . . . , k. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums.