2025 Putnam A1

Let $m_0$ and $n_0$ be distinct positive integers. For every positive integer $k$,

define $m_k$ and $n_k$ to be the relatively prime positive integers such that

$$

\frac{m_k}{n_k} = \frac{2m_{k-1} + 1}{2n_{k-1}+1}.

$$

Prove that $2m_k+1$ and $2n_k+1$ are relatively prime for all but finitely many positive integers $k$.

令 $m_0$ 和 $n_0$ 为不同的正整数。对于每个正整数 $k$，

将 $m_k$ 和 $n_k$ 定义为互素正整数，使得

$$

\frac{m_k}{n_k} = \frac{2m_{k-1} + 1}{2n_{k-1}+1}。

$$

证明 $2m_k+1$ 和 $2n_k+1$ 对于除有限多个正整数 $k$ 之外的所有正整数而言都是互质的。