一类不定方程的解

A class of Diophantine equation

讨论不定方程ax^2+by^2+cz^2=m+dxyz。利用二元二次型和初等数学的知识,给出a,b,c都整除d时,方程存在基础解时正整数a,b,c,d和非负整数m所有可能的取值。对每一个有基础解的方程,求解得出它的基础解,由这些基础解可以计算得到方程的多个整数解。

A class of Diophantine equation ax^2+by^2+cz^2=m+dxyz was considered.Based on the definition of the fundamental solution,with the help of the binary quadratic forms and the fundemental mathmatical knowledge,all of the possibilities of positive integers a,b,c,d and nonnegative integer m in the equation could be figured out when ald,bld and cld.And meanwhile,the corresponding fundamental solution was computed in each case.By these fundamental solutions,many integer solutions of every equation can be derived.