Question 97
a) Show that, for every positive integer $n$, there exists uniquely determined positive integers $x_n, y_n$ such that $(1 + \sqrt{33})^n = x_n + y_n\sqrt{33}$.
b) Prove that if $x_n, y_n$ are defined as above and $p$ is a positive prime, then at least one of the numbers $y_{p-1}, y_p, y_{p+1}$ is divisible by $p$.
b) Prove that if $x_n, y_n$ are defined as above and $p$ is a positive prime, then at least one of the numbers $y_{p-1}, y_p, y_{p+1}$ is divisible by $p$.