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

Abstract

Let k, l >= 2 be fixed integers. In this paper, firstly, we prove that all solutions of the equation (x + 1)(k) + (x + 2)(k) + . . . + (lx)(k) = y(n) in integers x,y,n with x, y >= 1, n >= 2 satisfy n < C-1, where C-1 = C-1(l, k) is an effectively computable constant. Secondly, we prove that all solutions of this equation in integers x, y, n with x,y >= 1,n >= 2, k not equal 3 and I 0 (mod 2) satisfy max{x, y, n} < C-2, where C-2 is an effectively computable constant depending only on k and I.