The Discussion Of The Integer Solution On Several Classes Of Diophantine Equation

The search of Diophantine is very actively by mangy people, and the research achievements are widely used on the research field of other subjects, such as discrete mathematics, physics and economics. So there are still many people who have great interested in Diophantine equation. Rich in its content and no unified approach contribute to the difficulty of its solvability. In general, we can only afford some solving principles which combine the elementary and advanced methods to transform the Diophantine to some equations easy to deal with or with well-know results.We are familiar with the method and solution of the simple Diophantine equation and quadratic Diophantine equation. But for the solution of cubic Diophantine equation and high order Diophantine equation, there is no general conclusion, so it needs further discussing. The main contents of this paper are:First, it is showed in this paper the concept of Diophantine equation, recent research and approaches to deal with it.Second, prior knowledge to introduce congruence theory.Third, using the elementary, algebraic number theory and the theory of congruence, the integer solution of some Diophantine equation are discussed.1. By using the elementary method and algebraic number theory, the integer solution of the Diophantine equation x2+4=y1is discussed.2. Let p be an odd prime, using the elementary method and algebraic number theory, it is discussed that if p=27s2+1, where s is an even integer then the Diophantine equation, x3-1=py2, has only one solution (x,y)=(1,0) 3.This study has proved by the elementary method that the Diophantine equation nx(x+1)(x+2)(x+3)has no positive integer solution(x,y),when n=Q2p2k,p is an odd prime number,both Q and k are positive integers,while4+Q and Q is an even.4.With n as an odd number and n≥3, using Pell equations to prove that Indeter-minate equation(2n+1)x+[2n(n+1)]y=[2n(n+1)+1]zhas only positive integral solution x=y=Z=2.