HMMT 2017 T8

Does there exist an irrational number $\alpha>1$ such that

$\left\lfloor\alpha^{n}\right\rfloor \equiv 0 (\bmod 2017)$

for all integers $n \geq 1$ ?

Dataset text NuminaMath olympiads. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.