Question 183
Let $a, b, c, \alpha, \beta$ be the given integers and the sequence $(u_n)$ is defined by $u_1 = \alpha, u_2 = \beta, u_{n+2} = a u_{n+1} + b u_n + c$ for all $n \ge 1$.
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 exactly one of the following two statements is true:
i) There are infinitely many positive integers $m$, such that $u_{n_0}u_{n_0+1}\dots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$;
ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\dots u_{n_0+k} - 1$ is divisible by 2023.
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 exactly one of the following two statements is true:
i) There are infinitely many positive integers $m$, such that $u_{n_0}u_{n_0+1}\dots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$;
ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\dots u_{n_0+k} - 1$ is divisible by 2023.