Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
Problem
Source:
Tags: Recursive Sequences
TTsphn
21.10.2007 04:43
An lovely problem : Let consider the sequence as follow : 1$ x_1,..,x_{2001}$ where $ 0<|x_i-x_j|\leq 2001$ and $ x_i\in \{a_n\}$ 2$ (x_1)!+x_1,...,x_{2002}!+x_2002$ ... ... 2002${ \prod_{i=1}^{2001}{x_{n_i}}+x_{n_i}}$ From this relative we has $ x_{ki}|x_{mi}\forall m\geq k$ Apply pegeonhole we has exist ${ (x_{ki},x_{kj})\in\{a_n}$ So we has $ x_{ki}|x_{mi}$
Peter
21.10.2007 13:04
How exactly are you defining that sequence? I don't see the pattern.
parsa1999
21.10.2014 17:37
so, nobody know the solution?
lazizbek42
05.10.2022 15:20
is there somobody who know the solution?