Problem

Source: China TST 2007, P2

Tags: floor function, number theory unsolved, number theory



A rational number $ x$ is called good if it satisfies: $ x=\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)=1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}-\alpha|\leq\dfrac{1}{2(p+q)}\] Find all the good numbers.