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

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).