298.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 298

References:

erdosproblems.com/298
[Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).

Imports (1)

FormalConjecturesUtil

theorem Erdos298.erdos_298 Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$? sorry line 45 source
Source documentation
Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$?
The answer is yes, proved by Bloom [Bl21] (even if 'positive density' is interpreted as 'positive upper density', which is likely what Erdős intended).
The theorem below uses the positive-upper-density interpretation; the literal natural-density interpretation is recorded separately.
This was formalized in Lean 3 by Bloom and Mehta.

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Not, Membership.mem, OfNat.ofNat, LT.lt, Real, Set.upperDensity, Set.univ, Exists, Finset, And, LE.le, Eq, Rat, Finset.sum, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_298 : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity →
    ∃ (S : Finset ℕ), ↑S ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := by
…
```
theorem Erdos298.erdos_298.variants.natural_density The literal natural-density interpretation of Erdős Problem 298 follows from [Bl21] sorry line 53 source
Source documentation
The literal natural-density interpretation of Erdős Problem 298 follows from [Bl21].

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Not, Membership.mem, OfNat.ofNat, Set.HasPosDensity, Set.univ, Exists, Finset, And, LE.le, Eq, Rat, Finset.sum, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_298.variants.natural_density : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A →
    A.HasPosDensity →
    ∃ (S : Finset ℕ), ↑S ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := by
…
```