On The Je(?)manowicz Conjecture Of Pythagorean Triples

An exponential indefinite equation ax +by =cz,in which a,b,c are positive integers,especially when they are quotient height array,is the most typical class of indefinite equations in the field of numerical theory.Because of its close relationship with coding theory,group theory and combinatorial theory,this kind of uncertain equation has always been popular among mathematical lovers.In this paper,we discuss some special cases of the conjectrue of Jesmanowicz concerning pythagorean triples for the Diophantine equation(a2-b2)x +(2ab)y=(a2 +b2)z.Based on the theory of simple congruent,second residual and fourth residual characteristics,get the following conclusion.On the situation of a-b = m(m ? 7(mod8)),b = n and m| n,when one of the following conditions is established,the Jesmanowicz conjecture.(?)n ? 3(mod4),and have prime number p?5(mod 8)make p |(2n(n + m)+ m2)and(2n+m/p)=1;(?)n ?0(modl6);(?)n ? 9(mod 16),and have prime number p ? 5(mod8)make p |(2n + m);(?)n ? 2(mod4),and 2n + m without shape factor of 4k+1;In particular,this conclusion holds up when m =7.On the situation of a-b = m(m = 3(mod 8)),b = n,and m|n,when one of the following conditions is established,the Jesmanowicz conjecture.(?)n ? 3(mod 8),m ? 3(mod16),and 2n +m has shape factor of p(?)1(mod 16);(?)n ? 7(mod 8),m?11(mod 16),and 2n + m has shape factor of p(?)4(mod 16);(?)n ? 9(modl6),m ? 3(mod8);(?)n ?1(mod 16),m ? 3(mod 8),and 2n+m without shape factor of 4k +1;On the situation of a-b=m(m ? 5(mod 8)),b=n,and,m| n,when one of the following conditions is established,the Jesmanowicz conjecture.(?)n?2(mod8),m?5(mod16),and 2n +m has shape factor of p(?)1(mod 16);(?)n?6(mod8),m ? 13(mod 16),and 2n+m has shape factor of p(?)1(mod 16);(?)n ? 4(mod16),m = 5(mod16);...