What can I say of x given these conditions? I know that $x^2-x$ is integer and also $x^n-x$ is integer for some $ n > 2$. Can I say that $x$ is integer? How can I show it? ($x \in \mathbb{R}$)

False, e.g. a root of $\,x^2\!-\!x\!+\!1\mid x^3\!+\!1\mid x^6\!-\!1\mid x^7\!-\!x\,$ has $\,x^2\!-\!x = -1,\ x^7\!-\!x = 0\, $ but $\,x\not\in\Bbb Z$
