| Pell's equation is one of the oldest problems studied in number theory, as the classic representation of the quadratic Diophantine equation, Pell's equation has always been concerned by scholars specialized in number theory, particularly relating to research x2 -Dy2 =±1, and many have made considerable achievements, some of these results for solving the existence of solutions of some Diophantine equations are very helpful. However, solving x2 -Dy2 =±1 amounts to solving the minimum solution, but it is very difficult. Whether by trial or even scores of law, are often encountered in the long calculation, and only to solve specific values of D. Therefore, looking for the simple methods for solving the minimum solution and exploring the application value of the Pell's equation are important subjects in number theory.The main achievements contained in this dissertation are as follows:1 Using the elementary method to discussed the relationship of the minimum solutions between the Pell's equation x2-Dy2 =1 and x2 -D1y2 =1(D = d2D1) where d = 2,3,5.2 Using some results of the Pell's equations, the existence of solutions of two classes of cubic Diophantine equations is discussed. Several sufficient conditions under which the Diophantine equations have no positive integer solution are given. |
