On a generalization of a problem of Erdős and Graham
Let \(f(x,k,d)\) be the product of \(k\) terms in an arithmetic progression of difference \(d\) and first term \(x\). \textit{P. Erdős} and \textit{J. L. Selfridge} [Ill. J. Math. 19, 292--301 (1975; Zbl 0295.10017)] proved that \(f(x,k,1)\) is never a perfect power of exponent \(\geq 2\) if \(x\geq 1\) and \(k\geq 2\) are integers. Their result spurred a lot of activity especially by Shorey and his collaborators on Diophantine equations involving perfect powers in \(f(x,k,d)\). In the paper under review, the authors study for fixed distinct integers \(a,b\) the integers \(x\) such that \(x(x+1)(x+2)(x+3)/(x+a)(x+b)\) is a square of a rational number. The main observation is instead of dividing out \((x+a)(x+b)\) from \(x(x+1)(x+2)(x+3)\) and getting a square of a rational, one might as well multiply the two expressions getting a square of an integer. The resulting equation is of the form \(f(x)=y^2\), where \(f\) is a monic polynomial with integer coefficients of degree \(6\), so one can use Runge's method to bound the integer solutions. The author's main results are explicit upper bounds on \(| x| \) in terms of \(a\) and \(b\), and a complete list of solutions for all distinct \(a,b\in \{-4,-3,-2.-1,4,5,6,7\}\).
scientific article; zbMATH DE number 6330564