On Prielippʹs Problem on Signed Sums ofkth Powers Original Research Article

In this paper we consider the representation of and integer,n, in the form[formula]wheremis a positive integer, depending onnand thevar epsiloniare all either ± the particular choices of thevar epsilonidepending onnandm. Among other things we prove such a representation always exists; in fact, infinitely many exist. The proof is algorithmic, so that a polynomial time method of finding the representation is presented. We get asymptotic estimates for the minimal value ofm, in each of the cases (a) kfixed andngrows to infinity and (b) nfixed andkgrows to infinity. A number of related problems and conjectures are also presented.