关于指数Diophantine方程x^2+b^y=c^z(英文)

On the exponential diophantine equation x^2+b^y=c^z

设m是偶数,r是奇数.设Ur,Vr是适合Vr+Ur-1=(m+-1)r的整数.证明了:当a=|Vr,|b=|Ur,|c=m2+1,r≡3(mod 4),m>r/π且m是2的方幂时,方程x2+by=cz仅有正整数解(x,y,z)=(a,2,r).

Let m be an even integer, and let r be an odd integer. Let Ur, Vr be integers satisfying Vr＋Ur √-1=（m＋√-1）^r. It is shown that if a=│Vr│, b=│Ur│, c=m^2＋1, r≡3（mod 4） and m is a power of 2 with m〉r/π, then the equation x^2＋b^y=c^z has only the positive integer solution（x, y, z）=（a, 2, r）.