Diophantine Equations With Applications In Mathematical Competition

In this paper,some methods for solving the diophantine equation are introduced through examples.The application of the diophantine equation in the mathematical competition is demonstrated through studying two problems of mathematical competition.In chapter 1,we briefly introduce the research background,the purpose of research,research significance,research contents and research method of the issue about diophantine equation and its application in mathematic contest.In chapter 2,we introduce some elementary methods for solving the diophantine equation.The methods include congruence method,factorization method,inequality method,infinite descent method,comparative prime power method,structure method,quadratic residue method,and Pell equation method.In chapter 3,we introduce two advanced methods for solving the diophantine equation.The methods are unique factorization domain method and cubic residue method.In chapter 4,the application of the diophantine equation in the mathematical competition is demonstrated through studying two problems of mathematical competition.One of the problems is solved by the congruence theory and we get a few theorems about congruence theory.The other problem is concentrating on solving the topic raised by Teacher KangHua Xu on his WeChat account.The we set a few problems of mathematical competition.In Chapter 5,we summarize the main content of the full text and propose some deficiencies in this paper.