Squares from blocks of consecutive integers: a problem of Erdős and Graham
Let \(n, k\in\mathbb{N}\) and put \[ f(n,k)=n(n+1)\cdots(n+k-1). \] An old result of \textit{P. Erdős} and \textit{J. Selfridge} [Ill. J. Math. 19, 292--301 (1975; Zbl 0295.10017)] states that the \(f(n,k)\) is never a perfect power of an integer, provided \(n\geq 1\) and \(k\geq 2.\) \textit{P. Erdős} and \textit{R. Graham} [Old and new problems and results in combinatorial number theory. Gene\`{v}e: L'Enseignement Mathématique, Université de Gene\`{v}e (1980; Zbl 0434.10001)] asked whether a more general Diophantine equation \[ \prod_{j=1}^{r}f(n_{j},k_{j})=x^2 \tag{1} \] could have, for fixed \(r\geq 1\) and \(\{k_{1},k_{2},\ldots,k_{r}\}\) with \(k_{j}\geq 4\) for \(j=1,2,\ldots,r\), at most finitely many solutions in positive integers \((n_{1},n_{2},\ldots,n_{r},x)\) satisfying the condition \[ n_{j}+k_{j}\leq n_{j+1}\tag{2} \] for each \(j\in\{1,\ldots,r-1\}\). The last condition guarantees that the corresponding blocks of integers are disjoint. The authors concentrate on the case \(k_{1}=k_{2}=\ldots=k_{r}=5\) and prove that if \(r\geq 5\) then there are infinitely many integers solutions of equation (1) satisfying (2), and thus answering the question of Erdős and Graham in the negative.
scientific article; zbMATH DE number 6009292
scientific article