关于指数Diophantine方程a^x+b^y=c^z的一个猜想(英文)

A conjecture concerning the exponential Diophantine equation ax+by=cz

设r是大于1的正奇数,m是偶数。设U_r,V_r是适合V_r+U_r(-1)^(1/2)/=(m+(-1)^(1/2))~r的整数,又设a=│V_r│,b=│U_r│,c=m^2+1。证明了:当a≡2(mod 4),b≡3(mod 4);m≥41r^(3/2)时,方程a^x+b^y=c^2仅有正整数解(x,y,z)=(2,2,r)。

Let r be an odd integer with r>1, and let m be an even integer. Let a=|Vr|,b=|Ur|, c=m2+1, where Ur, Vr are integers satisfying V,+ Ur-1 = (m+-1)r. It will be prowed that if a=2(mod 4), b=3(mod 4) and m>41r3/2, then the equation ax+by=cz has only the positive integer solution (x,y,z)=(2, 2, r).