广义商高数的纯指数Diophantine方程a^x+b^y=c^z

The Pure Exponential Diophantine Equation a^x + b^y = c^z for Generalized Pythagorean Triplets

设r是大于1的正奇数,m是正偶数;又设(a,b,c)=(|V(m,r)|,|U(m,r)|,m^2+1),其中.本文运用Gel'fond-Baker方法证明了:当m>10~6r^6。时,方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,r).

Let r be a positive odd integer with r 1,and let m be a positive even integer.Further more let {a,b,c） =（｜V（m,r）｜,｜U（m,r）｜,m^2 ＋ 1）,where .In this paper,using the Gel＇fond-Baker method,we prove that if m10~6r^6,then the equation a^x＋b^y=c^z has only the positive integer solution （x,y,z） = {2,2,r）.