2004 IMO Shortlist S12

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$ . Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by $$x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,$$ has all of its terms relatively prime to $m$ .

*Proposed by Jaroslaw Wroblewski, Poland*

令 $k$ 为大于 1 的固定整数，并令 ${m=4k^2-5}$ 。证明存在正整数 $a$ 和 $b$，使得由 $$x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,$$ 定义的序列 $(x_n)$ 的所有项都与 $m$ 互质。

*由波兰 Jaroslaw Wroblewski 提出*