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

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$$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$$TheabovecongruenceallowsustostateanequivalentformulationofGiug{a}^{\prime }sconjecture.Moreover,weprovethatthefirstabovecongruenceissatisfiedmodulo$n^4$whenever$n\ge 5$isaprimenumbersuchthat$n-1\nmid 2k-2$.Inparticular,thiscongruencearisesaconjectureforaprimetobeWolstenholmeprime.WealsoproposeseveralGiuga-Ago{h}^{\prime }s-likeconjectures.Further,weestablishtwocongruencesmodulo$n^3$fortwobinomialtypesumsinvolvingsumsofpowers$S_{2i}(n)$with$i=0,1,...,k$.Furthermore,usingtheabovecongruencereducedmodulo$n^2$, we obtain an extension of Carlitz-von Staudt result for odd power sums.

16 pages; the manuscript contains 7 new Giuga-Agoh's-like conjectures

References in corpus (2)

• The Equivalence of Giuga's and Agoh's Conjectures
• Generalizing Giuga's conjecture