Discussion And Research About The Integer Solutions Of Diophantine Equations

Diophantine equation is an important subject in number theory. It was greatly connected with algebra, combinatorial mathematics and computer science etc. The achievements in diophantine equations play an important role both in every branch of mathematics and in other subjects, such as physics, economics. So there are still many people who have great interested in diophantine equations.In 1999, the diophantine equation2x~2 + 2y~2 + 3z~2 = 1 + 6xyzappeared in connection with the description of the lower part of the approximation spectrum for quaternion, which inspire us to study such kind of diophantine equationax2 + by2 + cz2 = m + dxyzfor the convenience of the application in other fields.On conditions that a, b, c, d are all positive integers and m is a non-negative integer, and a | d, b | d, c | d, the main results of this paper is firstly giving the methods to solve the equation ax2 + by2 + cz2 = m + dxyz, the congruent method and the method of the Pell's equation.Next, I find (a,b,c,d,m), which make the equation ax2 + by2 + cz2 = m + dxyz has some fundamental solution(s), and all the corresponding fundamental solutions given in different tabulars. Then every integer solution to the equation comes through the neighbouring process from a unique fundamental solution.Finally, the integer solutions satisfying some conditions to the diophantine equationax2 + by2 + cz2 — x + dxyz were given.