四元四次不定方程x~4+y~4+(x+y)~4=z~4+w~4+(z+w)~4 (1)除了 x=z,y=w 或 x=w,y=z 时的平凡解以外,是否有非平凡解呢?我们可以“凑出”一组:1~4+9~4+(1+9)~4=5~4+6~4+(5+6)~4事实上,我们可用文[1]的方法来求(1)的通解:先将(1)两边析因:2(x~2+xy+y~2)~2=2(z~2+zw+w~2)~2因 x~2+xy+y~2≥0,z~2+zw+w~2≥0,故有x~2+xy+y~2=z~2+zw+w~2 (2) Quaternary four indefinite equations x~4+y~4+(x+y)~4=z~4+w~4+(z+w)~4 (1) except x=z, y=w, or x Is there a non-trivial solution other than the ordinary solution at =w,y=z? We can “make up” a group: 1~4+9~4+(1+9)~4=5~4+ 6~4+(5+6)~4 In fact, we can use the method of [1] to find the general solution of (1): First, (1) Both sides of the factor: 2 (x~2+xy+y~2) )~2=2(z~2+zw+w~2)~2 Since x~2+xy+y~2≥0, z~2+zw+w~2≥0, there is x~2+xy +y~2=z~2+zw+w~2 (2)