On The Exponential Diophantine Equation A~x+b~y=c~z

In this paper,we prove that the Diophantine equation ax-by=cz holds in a special case c=2pq. Base on the method of simple congruence,comparative the prime factors,factorization and quadratic residue.Theorem 1.Assuming that p,q are prime numbers and p (?) ab,q(?)ab,the Diophantine equation ax-by=(2pq)z has no positive integer solution except a=2(pq)4+1,b=2(Pq)4-1 and a= 4+(Pq)4,b=(pq)4-4.The positive integer solutions based on p≡3(mod 4), g≡3(mod 4).