FormalConjectures/ErdosProblems/853.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 853

Reference: erdosproblems.com/853

Imports (1)
  1. def Erdos853.r Let r(x) be the smallest even integer t such that primeGap = t has no solutions for n ≤ x line 36 source

    Source comment

    Let `r(x)` be the smallest even integer `t` such that `primeGap = t` has no solutions for `n ≤ x`.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Eq, HMod.hMod, Not, Exists, LE.le, primeGap

    6 instances, coercions and auxiliary constants left out.

    noncomputable def r (x : ℕ) : ℕ :=
…
  2. theorem Erdos853.erdos_853.parts.i Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime sorry line 46 source

    Source documentation

    Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$.

    Is it true that $r(x) \to \infty$?

    depends on sorry what this means

    Statement uses: Filter.Tendsto, Nat, Erdos853.r, Filter.atTop

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

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_853.parts.i : atTop.Tendsto r atTop := by
…

    Mentions: Erdos853.r

  3. theorem Erdos853.erdos_853.parts.ii Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime sorry line 55 source

    Source documentation

    Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$.

    Is it true that $r(x) / \log x \to \infty$?

    depends on sorry what this means

    Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos853.r, Real.log, Filter.atTop

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

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_853.parts.ii :
    atTop.Tendsto (fun n ↦ r n / Real.log n) atTop := by
…

    Mentions: Erdos853.r