On the equation \(s(1^ k+2^ k+\cdot \cdot \cdot +x^ k)+r=by^ z\) (Q2641328)

Consider the title equation, where r,s,b, and k are given nonzero integers \((k>0)\). In this paper new conditions on r,s, and k are established under which the equation has only finitely many solutions in integers \(x>0\), y with \(| y| \geq 2\) and \(z\geq 2\). These conditions can all be stated in terms of divisibility properties of the integer constants r,s, and k by powers of 2. The finiteness argument is provided by results of \textit{K. Györy}, \textit{R. Tijdeman} and \textit{M. Voorhoeve} [Acta Arith. 37, 233-240 (1980; Zbl 0365.10014)].