Question 83
Let $x$, $y$ and $a_{0}$, $a_{1}$, $a_{2}$, $\ldots$ be integers satisfying $a_{0}=a_{1}=0$ and
$$
a_{n+2}=x \cdot a_{n+1}+y \cdot a_{n}+1
$$
for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\operatorname{gcd}\left(a_{p}, a_{p+1}\right)$ is either equal to $1$ or greater than $\sqrt{p}$.
$$
a_{n+2}=x \cdot a_{n+1}+y \cdot a_{n}+1
$$
for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\operatorname{gcd}\left(a_{p}, a_{p+1}\right)$ is either equal to $1$ or greater than $\sqrt{p}$.