Question 34
Let $(x_n)$, $n \in \mathbb{N}^*$ be a sequence which is recursively defined by
$x_{n+1} = 3x_n^3 + x_n,$
where $x_1 = \frac{a}{b}$, and $a, b$ are positive integers such that $3$ doesn't divide $b$. If for some positive integer $m$ we have that $x_m$ is a perfect square of a rational, prove that $x_1$ is a perfect square of a rational.
$x_{n+1} = 3x_n^3 + x_n,$
where $x_1 = \frac{a}{b}$, and $a, b$ are positive integers such that $3$ doesn't divide $b$. If for some positive integer $m$ we have that $x_m$ is a perfect square of a rational, prove that $x_1$ is a perfect square of a rational.