Types Of Three Lost Diophantine Equations

The so-called Diophantine equation is the indeterminate equation in number theory in which the number of variable is more than that of the equation(or equations). Diophantine equation is an important subject in number theory and closely connected with algebraic number theory, combinatorics, algebraic geometry and computer science etc. The achievements in Diophantine equation play an important role both in every branch of mathematics and in other subjects, such as economics, physics. So there are still many people who have great interested in Diophantine equation.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, using the elementary and the theory of congruence I studied with the Diophantine equation in which an integer of the issue. In this part, we will prove that the equation x3-38y2=1 has integer solutions (x,y)=(1,0),(3,4)ï¼›the equation x3-73y2=1 has integer solutions (x,y)= (1,0); the equation x3-97y2=1 has integer solutions (x,y)= (1,0).Second, using some results of the Pell's equation, the existence of solutions of two classes of cubic Diophantine equation is discussed. Several sufficient conditions under which the Diophantine equation have no positive integer solution are given.