數學題--x^3 y^3 z^3之類題型一問

Question

x^3+y^3+z^3=x+y+z=3
(x,y,z)=?
有四組解 請各位幫幫忙 小弟非常感謝^^

Answer 1

x+y+z=3
z = 3 - x - y
x3+y3+z3
= x3+y3+(3-x-y)3
= -3x2y-3xy2+18xy+9x2+9y2-27x-27y+27 = 3
=> x2y+xy2 + 3(x2+2xy+y2) - 9(x+y) = 8
=> xy(x+y)+3(x+y)2-9(x+y) = 8
=> (x+y)(xy + 3x + 3y -9) = 8
=> (x+y)[x(y+3) -3(y+3)] = 8
=> (x+y)(x-3)(y-3) = 8
依 x+y, x-3, y-3 = +-1, +-2, +-4, +-8 的情形討論
得 x=y=1; x=y=4; x=4, y=-5; x=-5, y=4 是僅有的四組解
=> (x, y, z) = (1,1,1) or (4,4,-5) or (4,-5,4) or (-5,4,4)
如果有問題, 請來函討論. 不然, 我可能會錯失你再補充的疑點.


2007-01-31 10:03:18 補充：
忘了註明
本題需限制 x, y, z 均為整數
否則將有無限多解