On The Conjecture Of Jesmanowicz Concerning Pythagorean Triples

In this paper,we proves that the conjecture of Jesmanowicz concerning pythagor-can triples holds true in some special cases base on elementary congruence and quadratic residue.we main results of this paper are the following theorems:Let 3+n,if(3(2n+3))x+(2n(n+3))y=(2n(n+3)+9)z satisfies one of the following cases:(1)n≡2(mod4),(2)n≡4(mod16), the conjecture of Jesmanowicz is hold.Let 5+n,if(5(2n+5))x+(2n(n+5))y=(2n(n+5)+25)z satisfies one of the following cases:(1)n≡1(mod4),(2)n≡7(mol16), the conjecture of Jesmanowicz is hold.Let 5+n,if(m(2n+m))x+(2n(n+m))y=(2n(n+m)+m2)z satisfies one of the following cases:(1)n≡1(mod4),(2)n≡7(mod16), the conjecture of Jesmanowicz is hold.Finally,we discuss the Diophantine equation x3+1=201y2 and give its all the integer solutions.