487.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos487.erdos_487 Let $A\subseteq \mathbb{N}$ have positive density sorry line 50 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Set.HasPosDensity, Set.univ, Exists, And, Membership.mem, Ne, Eq, Nat.lcm
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: c