MathDB
purely NT problem with sequences not necessarily composed of numbers

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February 7, 2021
number theoryalgebraseuqencesPeriodicity

Problem Statement

Let be two real numbers α,β \alpha ,\beta and two sequences (xn)n≥1,(yn)n≥1 \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} whose smallest periods are p,q, p,q, respectively. Prove that the sequence (αxn+βyn)n≥1 \left( \alpha x_n+\beta y_n\right)_{n\ge 1} is periodic if gcd2(p,q)∣lcm(p,q), \text{gcd}^2 (p,q) | \text{lcm} (p,q) , and in this case find its smallest period.
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