##
**
On a Congruence Modulo ***n*^{3}
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 *k*th 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 *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* ≥ 5
is a prime such that
*n* -1 ∤ 2*k*-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, ..., *k*. Finally, we obtain an
extension of a result of Carlitz-von Staudt for odd power sums.

**
Full version: pdf,
dvi,
ps,
latex
**

(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.

Return to
**Journal of Integer Sequences home page**