关于丢番图方程x^2+b^y=c^z

ON THE DIOPHANTINE EQUATION x2+by=cz

设ｓ，ｔ为满足（ｓ，ｔ）＝１，ｓ＞ｔ的正整数，ａ＝２ｓｔ，ｂ＝ｓ２－ｔ２，ｃ＝ｓ２＋ｔ２．证明了：若ｃ为素数幂且ｂ≡±５（ｍｏｄ８），则不定方程ｘ２＋ｂｙ＝ｃｚ仅有一组正整数解（ｘ，ｙ，ｚ）＝（ａ，２，２）．

Let s,t be positive integers with (s,t)=1,s>t,a=2st,b=s2-t2,c=s2+t2.The author proved that:if c is a prime power and b≡±5(mod8),then the equation x2+by=cz has the only positive integral solution (x,y,z)=(a,2,2).