On the Diophantine Equation n^x + 10^y = z^2

Abstract

In this paper, we show that (n, x, y, z) = (2, 3, 0, 3) is the unique non-negative integer solution of the Diophantine equation n^x + 10^y = z^2 , where n is a positive integer with n ≡ 2 (mod 30) and x, y, z are non-negative integers. If n = 5, then the Diophantine equation has exactly one non-negative integer solution (x, y, z) = (3, 2, 15). We also give some conditions for non-existence of solutions of the Diophantine equation.