FormalConjectures/ErdosProblems/455.lean

GitHub source file

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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 455

References:

Imports (1)
  1. theorem Erdos455.erdos_455 Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n sorry line 36 source

    Source documentation

    Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?

    depends on sorry what this means

    Statement uses: Iff, True, Nat, StrictMono, And, Nat.Prime, GE.ge, HSub.hSub, HAdd.hAdd, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, HPow.hPow, Filter.atTop

    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_455: answer(sorry) ↔ ∀ q : ℕ → ℕ, StrictMono q →
    (∀ n, (q n).Prime ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
    Tendsto (fun n : ℕ => (q n : ℝ) / n ^ 2) atTop atTop := by
…
  2. theorem Erdos455.erdos_455.variants.liminf Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n sorry line 45 source

    Source documentation

    Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Then liminf q n / (n ^ 2) > 0.352, and this is proved in [Ri76].

    depends on sorry what this means

    Statement uses: Nat, StrictMono, And, Nat.Prime, GE.ge, HSub.hSub, HAdd.hAdd, OfNat.ofNat, GT.gt, ENNReal, Filter.liminf, HDiv.hDiv, HPow.hPow, Filter.atTop, OfScientific.ofScientific, Bool.true

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

    58 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_455.variants.liminf : ∀ q : ℕ → ℕ, StrictMono q →
    (∀ n, (q n).Prime ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
    liminf (fun n : ℕ => (q n : ℝ≥0∞) / n ^ 2) atTop > 0.352 := by
…