Are there only finitely many solutions to $i \prod (m _{i} 2 m _{i}) = j \prod (n _{j} 2 n _{j})$ with the $m_{i}, n_{j}$ distinct?

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number theory · solved · formalized (Lean) · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

No. In fact there are **infinitely many** such identities with all indices distinct.

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AMS 11 · solved (literature)

theorem erdos_397 :
    answer(False) ↔
      {(M, N) : Finset ℕ × Finset ℕ | Disjoint M N ∧
      ∏ i ∈ M, centralBinom i = ∏ j ∈ N, centralBinom j}.Finite

formal-conjectures/397.lean ↗

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