"orl\ wrote: (b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that \[ x^a + y^b = z^c. \] See it here http://www.mathlinks.ro/viewtopic.php?t=150638

My solution. Exist u, v, w such that: $ u + v = w$ and: $ u = \prod_i^n p_i^{\alpha_i}$; $ v = \prod_i^n p_i^{\beta_i}$; $ w = \prod_i^n p_i^{\gamma_i}$; By chinese remainder theorem, exist $ \delta_i, i = 1,..n$ such that: $ a\mid\delta_i + \alpha_i$ $ b\mid\delta_i + \beta_i$ $ c\mid\delta_i + \gamma_i$ Choose $ z = \prod_i^n p_i^{\delta_i}$, This problem proven