| The Diophantine equation is an important part in number theory, which doesn't only have close connections with the algebraic number theory, the combinatorial number theory and so on, but also play an important role in other science subjects. Therefore, it is necessary for us to study and solve some basic types of the solutions of the Diophantine equations.Now, We are familiar to the solution of the simple Diophantine equation and quadratic Diophantine equation, while with the solution of cubic Diophantine equation, there is no general conclusion, so it needs further discussing.In this paper, with the elementary and algebraic number theory, I discussed the Diophantine equations x3+A=By2, where B can be exactdivided by the prime number 6k+1, the main work is:1. Investigating the positive integer solution of the Diophantine equation x3±1= Dy2, where D is a square-free positive integer and can be exactdivided by the prime number 6k+1.2. Investigating the positive integer solution of the Diophantine equation x3±1= 3Dy2, where D is a square-free positive integer and can be exactdivided by the prime number 6k+1.3. Investigating the positive integer solution of the Diophantine equation x3±8= pD1y2, where D1 is a square-free positive integer and can be exactdivided by the prime number 6k+1.4. With the method of algebraic number theory, investigating the integer solution of the Diophantine equation y2=x3-14... |
