ON THE DIOPHANTINE EQUATION x2

Abstract

In this paper, we find all integer solutions (x, y, n, a, b, c) of the equation in the title for non-negative integers a, b and c under the condition that the integers x and y are relatively prime and n >= 3. The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.