Question 33
Let $a$, $b \gt 1$ be relatively prime positive integers. Define a sequence
$$
x_1 = a, \quad x_2 = b, \quad x_n = \frac{x_{n-1}^2 + x_{n-2}^2}{x_{n-1} + x_{n-2}} \quad \text{for } n \ge 3.
$$
Prove that $x_n$ is not an integer for $n \ge 3$. (Tonći Kokan)
$$
x_1 = a, \quad x_2 = b, \quad x_n = \frac{x_{n-1}^2 + x_{n-2}^2}{x_{n-1} + x_{n-2}} \quad \text{for } n \ge 3.
$$
Prove that $x_n$ is not an integer for $n \ge 3$. (Tonći Kokan)