On products of disjoint blocks of consecutive integers
By a block of \(l\geq 2\) consecutive integers it is meant, in this paper, a quantity of the form \(f(x)=(x+1)\cdots (x+l)\), where \(x\) is a positive integer. One block of \(l\) consecutive integers can never be a (non-zero) perfect square, as a consequence of a well-known result by \textit{P. Erdős} and \textit{J. L. Selfridge} [``The product of consecutive integers is never a power'', Ill. J. Math. 19, 292--301 (1975; Zbl 0295.10017)]. The generalization of this problem to products of \(k\) blocks of \(l\) consecutive integers (\(k\) fixed) leads to solving the equation \[ y^2=\prod_{i=1}^k f(x_i) \tag{1} \] in positive integers \(x_1,\dots,x_k\) satisfying \(x_i+l\leq x_{i+1}\) for \(i=1,\dots,k-1\). Note that the last condition ensures that the blocks \(f(x_1),\dots,f(x_k)\) are pairwise distinct. Some years after the above mentioned result Erdős and Graham conjectured that, for fixed \(k\) and any \(l\geq 4\), equation (1) has finitely many solutions. This conjecture is also mentioned as problem D17 of \textit{R. K. Guy}'s \textit{Unsolved Problems in Number Theory} [3rd ed., New York, NY: Springer-Verlag (2004; Zbl 1058.11001)]. In this paper the author disproves this conjecture by giving explicitly an infinite set of solutions to (1) when \(k=l=4\). He also shows that for a \(k\geq 6\) fixed and \(l=4\), equation (1) has infinitely many solutions.
scientific article; zbMATH DE number 5125712