求满足X^3+Y^3+Z^3=9且X+Y+Z=3的所有整数解

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求满足X^3+Y^3+Z^3=9且X+Y+Z=3的所有整数解
求满足X^3+Y^3+Z^3=9且X+Y+Z=3的所有整数解

求满足X^3+Y^3+Z^3=9且X+Y+Z=3的所有整数解
x^3+y^3+z^3=(x+y+z)(x^2+y^2+z^2-xy-yx-xz)+3xyz
9=3*1/2[(x-y)^2+(x-z)^2+(y-z)^2]+3xyz
(x-y)^2+(x-z)^2+(y-z)^2=6-2xyz
6以内的完全平方数有0,1,4
则(x-y)^2+(x-z)^2+(y-z)^2=1+1+4 xyz=0 x,y,z=1,2,0
0+0+4 xyz=1舍
1+0+4 xyz=0.5 舍
0+0+0 xyz=3 舍
所以,x=1 y=2 z=0
或x=1 y=0 z=2
或x=2 y=1 z=0
或x=2 y=0 z=1
或x=1 y=2 z=0
或X=1 y=0 z=2

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