关于Diophantine方程a^x+b^y=z^2

The Diophantine Equation a^x+b^y=z^2

设a=2~r,b=p^s,其中p是给定的奇素数,r和s是给定的正整数.运用有关三项Diophantine方程和广义Ramanujan-Nagell方程的结果,将方程a^x+~y=z^2的所有正整数解(x,y,z)进行了分类,从而得出了这些解的可有效计算的上界.

Let a = 2r and b = ps, where p is a fixed odd prime, r and s are fixed positive integers. In this paper, using certain results on the ternary diophantine equation and the generalized Ramanujan-Nagell equations, M1 positive integer solutions （x, y, z） of the equation ax ＋ by = z2 are classified. Thus, an effectively computable upper bound for the solutions is given.