Equal sums of like powers, both positive and negative
In the literature there are several results concerning the problem of finding two sets of integers \(x_1,\dots,x_s\) and \(y_1,\dots,y_s\) such that \(\sum_{i=1}^s x_i^r=\sum_{i=1}^s y_i^r\) for certain exponents \(r=k_1,\dots,k_n\). In the introduction the author mentions several related results; now we only recall the famous Tarry-Escott problem, where \(r=1,\dots,n\). The author takes up the scarcely investigated problem where the exponent \(r\) can take negative values, as well. More generally, the author considers so-called chains, i.e. the problem of finding not only two, but \(t\geq 2\) different sets of integers having the same sums of \(r\)-th powers, where \(r\) runs through a fixed finite set of (possibly both positive and negative) integers. Among other results, it is proved that for any fixed set of exponents, one can find an \(s\), and \(t\) sets of integers \(x_{i1},\dots,x_{it}\) \((i=1,\dots,s)\), forming a chain. In certain cases parametric and numerical solutions are also given.
scientific article; zbMATH DE number 5934867