On The Jesmanowicz Conjecture Of Pythagorean Triples

In this paper, we prove that the conjecture of Jesmanowicz concerning pythagore-an triples for the diophantine equation ax+by=cz holds true in a special cases. Base on elementary congruence, quadratic residue, bi-quadratic residue characters and factorization method.Theorem. For the pythagorean numbers a=n2-4, b=4n,c=n2+4,2(?)n the conjecture of Jesmanowicz holds when n+2exists a prime factor p such that p≠1(mod16).In particular,we have thatCorollary.For the pythagorean numbers,the conjecture of Jesmanowicz holds when n such that n≠-1(mod16).