Generalized balancing numbers (Q1042756)

A positive integer \(x\) is called a \((k,\ell)\)--balancing number for \(y\) (where \(x\leq y-2\)) if \[ 1^k+2^k+\cdots + (x-1)^k = (x+1)^\ell + \cdots+(y-1)^\ell , \] for fixed positive integers \(k\) and \(\ell\). In this paper, the authors prove some effective and ineffective finiteness statements for balancing numbers, using certain Baker-type Diophantine results and the Bilu-Tichy theorem, respectively. In particular, they prove the following Theorem. For any fixed integer \(k>1\) there are only finitely many pair \((y,\ell)\) for which the above equation has a solution.