On the Diophantine equation (x+1)(k) + (x+2)(k) + ... + (2x)(k) = y(n)

Abstract

In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T-k(x) = (x + 1)(k) + (x + 2)(k) + ... + (2x)(k). Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6]), we show that for 2 <= x <= 13, k >= 1,y >= 2 and n >= 3 the title equation has no solutions.