<s>[INST] Find all $x,y,z \in \mathbb{Q}.x^2+y^2+z^2+x+y+z=1$ Find all $x,y,z \in \mathbb{Q}$
where.$x^2+y^2+z^2+x+y+z=1$
 [/INST] Assume $(x,y,z)$ is sastifying the equation.
We can multiply $K=N^2(N \in \mathbb{Z}^+)$ to the both sides of equation.
And the both sides are integers.
Set $N=2^{h}s$,$s$ is odd.
Then $K((2x+1)^2+(2y+1)^2+(2z+1)^2)=7N^2$
The $LHS$ is sum of 3 squares.
From A production by Gauss, $a^2+b^2+c^2=m,(a,b,c,m,\alpha,t \in\mathbb{N})$$\Leftrightarrow m\ne 4^{\alpha}(8t+7)$
Because,$s^2 \equiv 1\pmod{8}\Rightarrow 7s^2 \equiv 7 \pmod{8}$
Which is impossible!
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