Primary Study On Several Special Types Of Diophantine Equations

Diophantine equation is an old problem, especially, it's more difficult to apply primary method to resolve the problems related to Diophantine equation. There are many solutions, which are very simple and also easy to understand, but hard to think out. Another name of indeterminate equation is Diophantine equation, it's an important branch of the number theory, and also the most active in the field of mathematics through the whole history. In 1980, Ke zhao and Sun qi, the famous mathematicians published the first monograph about Diophantine equation---Take the Diophantine Equation on the basis of the monograph, in 1987, Cao zhenfu finished the script which gave us the general summary and systematic study of Diophantine equation and published by press. In the recent ten years, not only Diophantine equation itself develops very actively, but also promotes the development of other fields.Primary number theory ever had a splendid history, many world-famous puzzles which were called the crown of mathematics such as the Conjecture of Goldbach, Fermat last theorem are all primary number theory problem., then there were a long dreary time, because some people thought it the noble among mathematics, elegant but useless, others even took it as thinking gym for granted. However, with the rapid development of computer, the importance of primary number theory is increasingly shown. Now it has been applied to many fields widely such as computer science, combinatorial mathematics, signal of digital processing, so number theory has been hot program in the current field of mathematics.In this paper, we have a general knowledge about Diophantine equation throughthe study of linear Diophantine equation, Pell equation, x~3 + y~3 + z~3 = 3 ,the introduction of Fermat last theorem, and also some discussion about the Diophantine equation x~3 + y~3 + z~3 = 3 in Study of Diophantine equation written by Ke zhao andSun qi. Among it, the integer solution of Diophantine equation is still an unresolved problem. Among 15 famous problems of Diophantine equation written by Yang shichun, it was listed in. Among it, the author hoped to make some further and useful development, and it proved that there are not integer solution to some extent when two unknowns are equal. The solution of these problems helps the study of other Diophantine equations and also produces some references, and makes some contributions to the research of Diophantine equation system.