Authors: Haofeng Zhang
In this paper for equation Axm+Byn=Czk , where m,n,k > 2, x,y,z > 2, A,B,C≥1 and gcd(Ax,By,Cz)=1, the author proves there are no positive integer solutions for this equation using “Order reducing method for equations” that the author invented for solving high order equations, in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation under the assumption of no positive integer solutions for Ax^3+By^3=Cz^3 when Ax^m-i+By^n-i>Cz^k-i.
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