VMO 2023 Problem 2

Given are the integers $a , b , c, \alpha, \beta$ and the sequence $(u_n)$ is defined by $u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c$ for all $n \geq 1$ .

a) Prove that if $a = 3 , b= -2 , c = -1$ then there are infinitely many pairs of integers $(\alpha ; \beta)$ so that $u_{2023}=2^{2022}$ .

b) Prove that there exists a positive integer $n_0$ such that only one of the following two statements is true:

i) There are infinitely many positive integers $m$ , such that $u_{n_0}u_{n_0+1}\ldots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$ ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1$ is divisible by $2023$