MathDB

6

Part of 2019 Rioplatense Mathematical Olympiad, Level 3

Problems(1)

Sequence of integers

Source: Rioplatense Olympiad L3 2019

12/10/2019
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.
algebranumber theory