Does there exist an irrational number $\alpha>1$ such that \[ \left\lfloor\alpha^{n}\right\rfloor \equiv 0 \quad(\bmod 2017) \] for all integers $n \geq 1$ ?
Let $\alpha>1$ and $0<\beta<1$ be the roots of $x^{2}-4035 x+2017$. Then note that $\left\lfloor\alpha^{n}\right\rfloor=\alpha^{n}+\beta^{n}-1$. Let $x_{n}=\alpha^{n}+\beta^{n}$ for all nonnegative integers $n$. It's easy to verify that $x_{n}=4035 x_{n-1}-2017 x_{n-2} \equiv x_{n-1}$ $(\bmod 2017)$ so since $x_{1}=4035 \equiv 1(\bmod 2017)$ we have that $x_{n} \equiv 1(\bmod 2017)$ for all $n$. Thus $\alpha$ satisfies the problem. $\fbox{Yes}$.