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
Montenegro

Abstract:

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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