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Sequence of integers

Source: Rioplatense Olympiad L3 2019

December 10, 2019
algebranumber theory

Problem Statement

Let α>1\alpha>1 be a real number such that the sequence an=α⌊αn⌋−⌊αn+1⌋a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor, with n≥1n\geq 1, is periodic, that is, there is a positive integer pp such that an+p=ana_{n+p}=a_n for all nn. Prove that α\alpha is an integer.
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