Question 59
Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^{n}+1$ are also in $S$. Prove that $S$ contains an integer of the form
$$
\left(p_{1} p_{2} \ldots p_{2023}\right)^{2023}
$$
where $p_{1}, p_{2}, \ldots, p_{2023}$ are distinct prime numbers, all greater than 2023.
$$
\left(p_{1} p_{2} \ldots p_{2023}\right)^{2023}
$$
where $p_{1}, p_{2}, \ldots, p_{2023}$ are distinct prime numbers, all greater than 2023.