三元三次不定方程ax^(2)+by^(2)+cz^(2)=dxyz-1基础解的进一步研究

Further Study of the Fundamental Solutions of Ternary Cubic Diophantine Equation ax^(2)+by^(2)+cz^(2)=dxyz-1

关于三元三次不定方程的研究,是不定方程研究中的重要课题,有许多尚未解决的问题.讨论了不定方程ax^(2)+by^(2)+cz^(2)=dxyz-1的基础解,其中(a,b,c)=1,a,b,c均为d的因子.利用文献中的方法,运用二元二次型理论和初等数论的结果,求出了该不定方程的所有基础解.

The study of ternary cubic Diophantine equations is an important subject in the study of indeterminate equations.There are many unsolved problems.In this paper,we discuss the fundamental solutions of the Diophantine equations ax^(2)+by^(2)+cz^(2)=dxyz-1,where(a,b,c)=1,and a|d,b|d,c|d.By means of the methods in references,using the results of binary quadratic form theory and elementary number theory,all the basic solutions of the Diophantine equation are obtained.