Research On Several Kinds Of Diophantine Equation

Diophantine equation is the integer algebraic equation(or equations) in which the number of variable is more than the number of the equation. Diophantine equation is a very important content and research topic in number theory. It is closely connected with algebraic number theory, combinatorics, algebraic geometry. The achievements in Diophantine equation play an important role both in every branch of mathematics and in other subjects, such as physics, economics, computer science. so there are still many people who have great interested in Diophantine equation.The main contents of this paper are :1. The overview of Diophantine equation, the main achievements of Diophantine equation, The difficulty and principle of solving Diophantine equation are discoursed.2. The preliminary knowledge of the paper are given, including congruence theory,quadratic residue, Legendre symbol, Pell equation Some of the main definitions, nature,theorem.3. The research progress of Diophantine equation Ax2+B=yn is introduced. by using the method of algebraic number theory proved that the Diophantine equation Ax2+B=ynwhen(A,B, n) =(1,4,9) has no integer solution.4. The research progress of Diophantine equation x2-Dy4=N is introduced. by using the method of recurrent sequence, congruence, quadratic remainder proved that the Diophantine equation x2-Dy4=N when(D,N)=(3,397) has only the integer solution(x,y)=(20,1).5. The research progress of Diophantine equation ax4+bx2y2+cy4=dz2 is introduced, and using the method of Fermat’s infinite descent proved that the Diophantine equation ax4+bx2y2+cy4=dz2when(a,b,c,d)=(2,2,1,1) has no positive integer solution.