A conjecture concerning the pure exponential diophantine equation ax + by = cz

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 inverted iota r. In this paper we prove that if a equivalent to 2 (mod 4), b equivalent to 3 (mod 4), c > 3. 10(37) and r > 7200, then the equation a(x) + b(y) = c(z) only has the solution (x, y, z) = (2, 2, r).