A Conjecture Concerning the Pure Exponential Diophantine Equation a^x+b^y=c^z

Let a, b, c, r be fixed positive integers such that a 2 * b 2 = c r, min(a, b, c, r) > 1 and 2 r. In this paper we prove that if a 2 (mod 4), b 3 (mod 4), c > 3.1037 and r > 7200, then the equation a x * b y = c z only has the solution (x, y, z) = (2, 2, r).