On The Jesmanowicz Conjecture Of Pythagorean Triples

In this paper,we prove that’the conjeeture of Je§manowl’cz concerning pythagore— an triples for the diophantine equaltion(s2-t2).+(2st)y=(s2+t2)x holds in some special cases.By using elementary congruence,quadratic residue and bi.quadratic residue characters,we get the following results:For the case of s一t=3,t=n,and 3 n,the Conjecturre of Jexmanowicz holds when one 0f the following conditions is.satisfied.(I)n≡3(mod8)；(II)n≡7(mod8),and 2n+3 exists a prime factor p such that p≠1(modl6)；(III)n≡9(modl6)；(IV)n≡4(modl6)；(V)n≡0(mod8),and 2n+3 doesn’t have any prime factor p Such thalt p≡1(mod4)；(VI)礼≡5(mod8),and 2礼+3 doesn’t have any prime flactor p Such that p≡l(mod4)；(VII)n≡1(modl6),and 2n+3 doesn-t have any prime fa.ctor p Such that p≡1(mod4).For the case of s—t=5,t=钆,and 5 n,the conjeeture of Jesmanowicz holds when one of the following conditions is satisfied.(I)n≡2(mod8)；(II)n三6(mod8),and 2n+5 exists a prime factor p such that p≠1(modl6)；(III)n≡4(mod8)；(IV)n≡7(mod8).