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a_n \in {1.-1} is periodic if a_1/3 + a_2/3^2 + ... =p/ q

Source: Canada Repêchage 2023/4 CMOQR

March 25, 2024
number theoryPeriodic sequenceperiodic

Problem Statement

Let a1a_1, a2a_2, ... ... be a sequence of numbers, each either 11 or −1-1. Show that if a13+a232+...=pq\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q} for integers pp and qq such that 33 does not divide qq, then the sequence a1a_1, a2a_2, ... ... is periodic; that is, there is some positive integer nn such that ai=an+ia_i = a_{n+i} for i=1i = 1, 22,.......
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