MathDB

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}

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

Problem Statement