Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4

On a Congruence Modulo n3 Involving Two Consecutive Sums of Powers

Romeo Meštrović
Maritime Faculty
University of Montenegro
85330 Kotor


For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1k + 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.

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(Concerned with sequences A000146 A000217 A000367 A000928 A002445 A002997 A007850 A027641 A027642 A027762 A029875 A029876 A046094 A064538 A079618 A088164 A165908 A177783 A198391 A199767 A219540 A226365.)

Received July 25 2013; revised versions received October 14 2013; October 22 2013; August 5 2014. Published in Journal of Integer Sequences, August 5 2014.

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