For various positive integers k, the sums of kth powers of the first n positive integers, S_k(n) := 1^k + 2^k + ... + n^k, are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo n³ 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 ≥ 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 S_2i(n) with i = 0, 1, ..., k. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums.