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Another 'good' number problem

Source: China TST 2007, P2

December 29, 2008
floor functionnumber theory unsolvednumber theory

Problem Statement

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