关于指数丢番图方程a^x+b^y=c^z

On the Exponential Diophantine Equation a^x+b^y=c^z

设r是大于1的奇数,m是偶数,Ur和Vr是适合Vr+Ur(-1)^(1/2)=(m+(-1)^(1/2))r的整数。运用初等方法,证明了:如果a=|Vr|,b=|Ur|,c=m2+1且b是素数,r≡3(mod4),m≡2(mod4),m>πr,那么方程ax+by=cz仅有正整数解(x,y,z)=(2,2,r)。

Let r be an odd integer with r＆gt;1,m be an even integer.Let Ur,Vr be integers satisfying Vr＋Ur√-1=（m＋√-1）r.By making use of elementary method,it is proved that if a=｜Vr｜,b=｜Ur｜,c=m^2＋1,r≡3（mod4）,m≡2（mod 4）,m〉r/π and b is a prime,then the equation a^x＋b^y=c^z has only the positive integer solution（x,y,z）=（2,2,r）.