FormalConjectures/ErdosProblems/299.lean

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L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 299

References:

Imports (1)
  1. theorem Erdos299.erdos_299 Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1? sorry line 45 source

    Source documentation

    Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?

    There does not exist such a sequence, which follows from the positive solution to [erdosproblems.com/298] by Bloom [Bl21].

    This was formalized in Lean 3 by Bloom and Mehta.

    depends on sorry what this means

    Statement uses: Iff, False, Exists, Nat, And, StrictMono, LT.lt, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, HSub.hSub, HAdd.hAdd, Finset, Ne, Finset.sum, HDiv.hDiv

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    34 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_299 : answer(False) ↔ (∃ (a : ℕ → ℕ),
    StrictMono a ∧ (∀ n, 0 < a n) ∧
    (fun n ↦ (a (n + 1) : ℝ) - a n) =O[atTop] (1 : ℕ → ℝ) ∧
    ∀ S : Finset ℕ, ∑ i ∈ S, (1 : ℝ) / a i ≠ 1) := by
…
  2. theorem Erdos299.erdos_299.variants.density The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21] sorry line 60 source

    Source documentation

    The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21].

    The statement is as follows: If $A \subset \mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive density) then there is a finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$.

    depends on sorry what this means

    Statement uses: Set, Nat, Not, Membership.mem, OfNat.ofNat, LT.lt, Real, Set.upperDensity, Set.univ, Exists, Finset, And, LE.le, Eq, Finset.sum, HDiv.hDiv

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    40 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_299.variants.density : ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity →
    ∃ S : Finset ℕ, ↑S ⊆ A ∧ ∑ n ∈ S, (1 : ℝ) / n = 1 := by
…