Question 2
For each integer $n\ge0$, put
$$u_n=(2+\sqrt5)^n+(2-\sqrt5)^n.$$
a) Prove that $u_n$ is a positive integer for every $n\ge0$. What is the largest possible remainder when $u_n$ is divided by 24?
b) Find all ordered pairs of positive integers $(a,b)$ with $a,b\lt 500$ such that for every odd $n$,
$$u_n\equiv a^n-b^n\pmod{1111}.$$
$$u_n=(2+\sqrt5)^n+(2-\sqrt5)^n.$$
a) Prove that $u_n$ is a positive integer for every $n\ge0$. What is the largest possible remainder when $u_n$ is divided by 24?
b) Find all ordered pairs of positive integers $(a,b)$ with $a,b\lt 500$ such that for every odd $n$,
$$u_n\equiv a^n-b^n\pmod{1111}.$$