Diophantine方程p^x+q^y=z^2（英文）

The Diophantine equation p^x+q^y=z^2

设p和q是两个奇素数,且p<q.B.Sroysang证明了如果(p,q)=(7,19)或(7,31),则方程px+qy=z2没有正整数解(x,y,z).为了研究这个问题,运用初等方法和指数Diophantine方程的一些性质,证明了一个一般结果,即如果p+q≡2(mod4)和(q|p)=-1,则方程有唯一的正整数解(p,q,x,y,z)=(3,11,5,4,122),其中(q|p)表示Legendre符号.

Let pand q be two odd primes with pq.Recently,B.Sroysang proved that if（p,q）=（7,19）or（7,31）,then the equation px＋qy=z2 has no positive integer solutions（x,y,z）.In order to study this problem,by using the elementary number theory methods and the properties of some exponential Diophantine equation,ageneral result is proved that if p＋q≡2（mod4）and（q｜p）=-1,where（q｜p）denotes the Legendre symbol,then the equation has only the positive integer solution（p,q,x,y,z）=（3,11,5,4,122）.