On the exponential diophantine equation $x^y+y^x=z^z$

For any positive integer D which is not a square, let (u 1, v 1) be the least positive integer solution of the Pell equation u 2 − Dv 2 = 1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2 l D and v 1h(4D) ≡ 0 (mod D), then D is called a singular number. In this paper, we prove that if (x, y, z) is a positive integer solution of the equation x y + y x = z z with 2 | z, then maximum max{x, y, z}