2010 CMO 第 3 题

Let $k$ be an integer $\geq 3$ . Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$ , we have

$$a_n =

\begin{cases}

a_{n-1}+1 & \text{if } (a_{n-1},n) = 1

2n & \text{if } (a_{n-1},n) > 1

\end{cases}

$$

Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$ .

令 $k$ 为整数 $\geq 3$ 。序列 $\{a_n\}$ 满足 $a_k = 2k$ 并且对于所有 $n > k$ ，我们有

$$a_n =

\开始{案例}

a_{n-1}+1 & \text{if } (a_{n-1},n) = 1

2n & \text{if } (a_{n-1},n) > 1

\结束{案例}

$$

证明序列 $\{a_n - a_{n-1}\}$ 中有无穷多个素数。