A congruence modulo $n^3$ involving two consecutive sums of powers and its applications

TL;DR

This paper proves new congruences involving sums of powers of integers, relates them to Giuga's and Wolstenholme's conjectures, and extends classical results on power sums with potential implications for prime number theory.

Contribution

It introduces novel congruences for sums of powers, connects them to longstanding conjectures, and extends classical theorems, offering new perspectives in number theory.

Findings

Proves a key congruence involving sums of powers modulo n^3.

Establishes conditions under which the congruence holds modulo n^4 for primes.

Proposes conjectures related to Giuga's and Wolstenholme's conjectures.

Abstract

For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\ge 2 $$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$$ The above congruence allows us to state an equivalent formulation of Giuga's conjecture. Moreover, we prove that the first above congruence is satisfied modulo $n^4$ whenever $n\ge 5$ is a prime number such that $n-1\nmid 2k-2$. In particular, this congruence arises a conjecture for a prime to be Wolstenholme prime. We also propose several Giuga-Agoh's-like conjectures. Further, we establish two congruences modulo $n^3$ for two binomial type sums involving sums of powers $S_{2i}(n)$ with $i=0,1,...,k$. Furthermore, using the above congruence reduced modulo $n^2$, we obtain an extension of Carlitz-von Staudt result for odd power sums.