On The Conjecture Of Jes'manowicz Concerning Pythagorean Triples

In this paper,using the method of simple congruence.quadratic residue.factorization.inequality in elementary number theory and biquadratic residue character theory in algebraic number the-ory,we prove that the conjecture of holds true for some special cases of the Diophantine equation (na)x+(nb)y=(nc)z.The main results of this paper are as follows:1.The Diophantine equation(n2-36)x+(12n)y=(n2+36)z has only the positive integer solution x=y=z=2.2?If the positive integer n satisfied n?3(mod20)or n?3(mod8),then the Diophan-tine equation (n(2n+7))x+(2n(n+7))y=(2n(n+7)+49)z has only the positive integer solution x=y=z=2.3.If the positive inte:ger n,a=72r?4 satisfied P(a)| nor P(n)|a,then the Diophantine equation (n(72r?4))x+(n(4.7r))y=(n(72r+4))z has only the positive integer solution x=y=z=2.