Question 2
Let $a,b,c,\alpha,\beta$ be integers and define $u_1=\alpha$, $u_2=\beta$, $u_{n+2}=au_{n+1}+bu_n+c$ for $n\ge1$.
a) Prove that for $a=3,b=-2,c=-1$, there are infinitely many integer pairs $(\alpha,\beta)$ such that $u_{2023}=2^{2022}$.
b) Prove that some positive integer $n_0$ has exactly one of the following properties:
i) There are infinitely many positive integers $m$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$.
ii) There are infinitely many positive integers $k$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+k}-1$ is divisible by 2023.
a) Prove that for $a=3,b=-2,c=-1$, there are infinitely many integer pairs $(\alpha,\beta)$ such that $u_{2023}=2^{2022}$.
b) Prove that some positive integer $n_0$ has exactly one of the following properties:
i) There are infinitely many positive integers $m$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$.
ii) There are infinitely many positive integers $k$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+k}-1$ is divisible by 2023.