Erdős Problem 730

References:

• erdosproblems.com/730
• A129515

theorem Erdos730.erdos_730 : S✝.Infinite ↔ sorry

Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

theorem Erdos730.erdos_730.variants.explicit_pairs : {(87, 88), (607, 608)} ⊆ S✝

For example, $(87,88)$ and $(607,608)$ are such pairs.

theorem Erdos730.erdos_730.variants.two_div_forall (n : ℕ) (h : 0 < n) : 2 ∣ (2 * n).choose n

Show that for all $n$, the binomial coefficient $\binom{2n}{n}$ is even.

theorem Erdos730.erdos_730.variants.delta_one (n m : ℕ) (h : (n, m) ∈ S✝) : m = n + 1

In every known example $(n, m) ∈ S$, we have $m = n + 1$.