On products of consecutive arithmetic progressions. III
In the paper the authors are interested in integer solutions of the Diophantine equation \[ y^2=\prod_{i=1}^{r}f(x_{i}, k_{i}, d), \] where \(f(x,k,d)=x(x+d)\cdots(x+(k-1)d)\). In other words, they are interested whether the products of certain blocks of integers forming an arithmetic progression can be a square. There are many results in the paper. For example, the authors prove that for \(r = 4, d=1\) and \(k_i=5\) for \(i=1, 2, 3, 4\), the equation under consideration has infinitely many positive integer solutions. Other results are obtained for various combinations of integers \(d, r\) and \(k_i\). The proofs are elementary and are based on finding suitable substitution which reduces the problem to a simpler equation (mainly of Pell type) or just by finding polynomial parametric solution. At the end of the paper many open question are formulated. For Part II see [the author, Acta Math. Hung. 156, No. 1, 240--254 (2018; Zbl 1424.11078)].
scientific article; zbMATH DE number 7399066