Summary: We go through a series of results related to the \(k\)-signum equation \(\pm 1^k\pm 2^k\pm\dots\pm n^k=0\). We are investigating the number \(S_k(n)\) of possible writings and the asymptotic behavior of these numbers, as \(k\) is fixed and \(n\to\infty\). The results are presented in connections with the Erdős-Surányi sequences. Analytic methods and algebraic ones are employed in order to predict the asymptotic behavior in general and to study in detail various situations for small values of \(k\). Some simplifications and further ramifications are discussed in the end about the recent proof of Andrica-Tomescu conjecture.
For the entire collection see [Zbl 1300.00029].