Squares From a Coupled System

Problem

Sequences $\{x_n\}$ and $\{y_n\}$ are defined by $x_1 = 1$ , $y_1 = 39$ , and

\begin{cases} x_{n+1} = 23x_n + y_n + 2, \\ y_{n+1} = 551x_n + 24y_n + 64. \end{cases}

Prove that $x_n$ is a perfect square for every positive integer $n$ .

Answer

Solution

Difficulty	8/10
Topics	number theory, sequences

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