Power values of sums of products of consecutive integers (Q2804246)

Consider the function NEWLINE\[NEWLINEf_k(x)=\sum_{i=0}^k \prod_{j=0}^i (x+j).NEWLINE\]NEWLINE The authors of the paper under review consider the Diophantine equation NEWLINE\[NEWLINEf_k(x)=y^nNEWLINE\]NEWLINE in integers \(x,y,k,n\) and prove several finiteness results. In particular, they show that there exist effectively computable upper bounds for (i) \(n\) provided that \(k\geq 1\) and \(y\neq 0,-1\), (ii) \(\max\{n,|x|,|y|\}\) provided that \(k\geq 1\) and \(n\geq 3\) and (iii) \(\max\{|x|,|y|\}\) provided that \(k\geq 2\) and \(n=2\). Moreover, they find all solutions \((x,y,n)\) for \(1\leq k\leq 10\) with \(k\neq 2\) if \(n=2\).