指数Diophantine方程a^x+b^y=c^z的例外解

The Exceptional Solutions to the Exponential Diophantine Equation a^x+b^y=c^z

设a,b,C是两两互素的正整数,min(a,b,C)>1.论文证明了:当b(?)1(mod 8),c(?)5(mod 8)且c是素数方幂时,如果ax+by=cz有正整数解(x,y,z)=(2,2,r),其中r是大于1的奇数,则该方程的例外解(x,y,z)都满足x=2以及y(?)z(?)1(mod 2).

Let a,b,c be coprime positive integers with min (a,b,c)>1. In this paper it is demonstrated that if b(?)1(mod8), c(?)5(mod8), c is a power of a prime and the equationax+by=cz has a positive integer solution (x, y, z) = (2,2, r), where r is an odd integer with r>1, then the exceptional solutions (x, y, z) of the equation satisfy x=2 and y=z=1(mod 2).