关于丢番图方程ax^4+by^4=cz^2

On the Diophantine equation ax^4+by^4=cz^2

目的对某类特殊的正整数a,b,c,寻找给出丢番图方程ax4+by4=cz2的全部正整数解的方法。方法利用初等方法把方程ax4+by4=cz2化为方程x2+my2=z2,给出方程ax4+by4=cz2的全部正整数解。结果给出了当(a,b,c)=(5,3,2)时方程ax4+by4=cz2的全部正整数解。结论利用上述方法可以解决一类方程ax4+by4=cz2的求解问题,从而拓展了Mordell等人关于ax4+by4=cz2的结果。

Aim To find the method of all positive integer solutions of Diophantine equation ax4＋by4=cz2 for some special positive integers a,b and c.Methods ax4＋by4=cz2 is changed into x2＋my2=z2 by using elementary methods,then all positive integer solutions of Diophantine equation ax4＋by4=cz2 are given.Results The all positive integer solutions of Diophantine equation ax4＋by4=cz2 are given when（a,b,c）=（5,3,2）.Conclusion The foregoing method may be used to solve a class of equations such as ax4＋by4=cz2 and the results of equation ax4＋by4=cz2 which is studied by Mordell and others are developed hereof.