Question 67
Consider the sequence
$$
\left\{\begin{array}{l}
x_{1}=2 ; \\
x_{n+1}=2 x_{n}^{2}-1 \quad \text{ for } n \geq 1
\end{array}\right.
$$
Prove that $n$ and $x_{n}$ are relatively prime for every $n \geq 1$.
$$
\left\{\begin{array}{l}
x_{1}=2 ; \\
x_{n+1}=2 x_{n}^{2}-1 \quad \text{ for } n \geq 1
\end{array}\right.
$$
Prove that $n$ and $x_{n}$ are relatively prime for every $n \geq 1$.