Question 170
An integer sequence $(x_n)$ is defined as follows: $0 \le x_0 \lt x_1 \le 100$ and
$$
x_{n+2} = 7x_{n+1} - x_n + 280, \forall n \ge 0.
$$
a. Prove that if $x_0 = 2, x_1 = 3$ then for each positive integer $n$, the sum of divisors of the following number is divisible by 24
$$
x_n x_{n+1} + x_{n+1} x_{n+2} + x_{n+2} x_{n+3} + 2018.
$$
b. Find all pairs $(x_0, x_1)$ such that $x_n x_{n+1} + 2019$ are perfect squares for infinitely many numbers $n$.
$$
x_{n+2} = 7x_{n+1} - x_n + 280, \forall n \ge 0.
$$
a. Prove that if $x_0 = 2, x_1 = 3$ then for each positive integer $n$, the sum of divisors of the following number is divisible by 24
$$
x_n x_{n+1} + x_{n+1} x_{n+2} + x_{n+2} x_{n+3} + 2018.
$$
b. Find all pairs $(x_0, x_1)$ such that $x_n x_{n+1} + 2019$ are perfect squares for infinitely many numbers $n$.