Question 92
Let $k$ be a positive integer and $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers such that
$$
a_{i} a_{i+1} \mid k-a_{i}^{2}
$$
for all integers $i \geq 1$. Prove that there exists a positive integer $M$ such that $a_{n}=a_{n+1}$ for all integers $n \geq M$.
$$
a_{i} a_{i+1} \mid k-a_{i}^{2}
$$
for all integers $i \geq 1$. Prove that there exists a positive integer $M$ such that $a_{n}=a_{n+1}$ for all integers $n \geq M$.