Power values of sums of certain products of consecutive integers and related results

Let n be a non-negative integer and put p(n) (x) = Pi(n)(i=0)(x + i). In the first part of the paper, for given n, we study the existence of integer solutions of the Diophantine equation y(m )= p(n)(x) + Sigma(k)(i=1)pa(i) (x), where m is an element of N->= 2 and a(1) < a(2 )< . . .< a(k) < n. This equation can be considered as a generalization of the Erdos-Selfridge Diophantine equation y(m) = p(n)(x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n >= 2 with a(1) >= 2, then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation y(m) = Sigma(k)(i=1) pa(i)(x(i)), for m = 2, 3, which can be seen as an additive version of the equation considered by Erdos and Graham. In particular, we prove that if m = 2, a(1) = 1 or m = 3, a(2) = 2, then for each k - 1 tuple of positive integers (a(2), . . . , a(k)) there are infinitely many solutions in integers. (C) 2018 Elsevier Inc. All rights reserved.