On The Conjecture Of Jes' Manowicz

The Diophantine equation not only developed actively itself, but also was applied to else fields of Discrete Mathematics. It plays an important role in people's study and Diophantine equation research to solve the actual problems. So many researchers study the Diophantine equation extensively and highly in the domestic and abroad. In this paper, the conjecture of Jes'manowicz is studied: as (a,b) = 1,a>b, 2|ab , the Diophantine equation (a~2 -b~2)~x+(2ab)~y = (a~2 +b~2)~z has only a positive integer solution (x, y, z) = (2,2,2), which is a very important problem in the exponential Diophantine equation. It has been several decade years ever since Jes'manowicz figured out the problem in 1956. During the period, many scholars have studied the problem.In this paper, minimizing constraint conditions, we receive six important theorems with the method of quadratic residue and quadratic reciprocity. We also prove that the conjecture is right for some special array, which pushes forward further the ultimate proof of the conjecture. The key to the establishment of the conjecture is to prove x, y and z are all even. This paper we will give a sufficient condition for 2|xy. In certain circumstances with a weakened a or a + b has 4k-1 shape factor of this constraint conditions that the suspect is established and on a≡0(mod 4),6≡1(mod 4) not under any restrictions into two types of special circumstances, proved this conjecture is established.In this paper, main result will be given in the third part.