Question 2
An integer sequence $(x_n)$ satisfies $0\le x_0\lt x_1\le100$ and
$$x_{n+2}=7x_{n+1}-x_n+280\qquad(n\ge0).$$
(a) If $x_0=2,x_1=3$, prove that for every natural number $n$, the sum of the positive divisors of $x_nx_{n+1}+x_{n+1}x_{n+2}+x_{n+2}x_{n+3}+2018$ is divisible by $24$.
(b) Find all pairs $(x_0,x_1)$ for which $x_nx_{n+1}+2019$ is a perfect square for infinitely many natural numbers $n$.
$$x_{n+2}=7x_{n+1}-x_n+280\qquad(n\ge0).$$
(a) If $x_0=2,x_1=3$, prove that for every natural number $n$, the sum of the positive divisors of $x_nx_{n+1}+x_{n+1}x_{n+2}+x_{n+2}x_{n+3}+2018$ is divisible by $24$.
(b) Find all pairs $(x_0,x_1)$ for which $x_nx_{n+1}+2019$ is a perfect square for infinitely many natural numbers $n$.