FormalConjectures.ErdosProblems.«487»

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import FormalConjecturesUtil

Erdős Problem 487

References:

erdosproblems.com/487

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

[DaEr36] Davenport, H. and Erdős, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.

[Kl71] Kleitman, Daniel, Collections of subsets containing no two sets and their union. Proceedings of the LA Meeting AMS (1971), 153-155.

namespace Erdos487

Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the least common multiple of $a$ and $b$)?

This is true, a consequence of the positive solution to [447] by Kleitman [Kl71].

Davenport and Erdős [DaEr36] showed that there must exist an infinite sequence $a_1<a_2\cdots$ in $A$ such that $a_i\mid a_j$ for all $i\leq j$, under the assumption that the upper logarithmic density of $A$ is positive.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos487.lean"]
theorem erdos_487 : answer(True) ↔
    ∀ A : Set ℕ, A.HasPosDensity →
      ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ Nat.lcm a b = c := by⊢ True ↔ ∀ (A : Set ℕ), A.HasPosDensity → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ a.lcm b = c
  sorryAll goals completed! 🐙

end Erdos487