FormalConjectures/ErdosProblems/695.lean

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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 695

Reference: erdosproblems.com/695

Imports (1)
Imported by 1 file
  1. theorem Erdos695.erdos_695 Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$ sorry line 39 source

    Source documentation

    Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$. Is it true that $$ \lim_{k \to \infty} q_k^{1/k} = \infty? $$

    depends on sorry what this means

    Statement uses: Iff, True, Nat, StrictMono, Nat.Prime, Eq, HMod.hMod, HAdd.hAdd, OfNat.ofNat, Filter.Tendsto, Real, HPow.hPow, HDiv.hDiv, Filter.atTop

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

    32 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_695 : answer(sorry) ↔
    ∀ {q : ℕ → ℕ},
      StrictMono q →
      (∀ i, (q i).Prime) →
      (∀ i, q (i + 1) % q i = 1) →
      Tendsto (fun k => (q k : ℝ) ^ (1 / k : ℝ)) atTop atTop := by
…
  2. theorem Erdos695.erdos_695.variants.upperBound Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $$ q(k) \leq \exp(k (\log k)^{1 + o(1)})? $$ sorry line 54 source

    Source documentation

    Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $$ q(k) \leq \exp(k (\log k)^{1 + o(1)})? $$

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Nat, And, StrictMono, Nat.Prime, Eq, HMod.hMod, HAdd.hAdd, OfNat.ofNat, Real, Asymptotics.IsLittleO, Filter.atTop, LE.le, Real.exp, HMul.hMul, HPow.hPow, Real.log

    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_695.variants.upperBound : answer(sorry) ↔
    ∃ q : ℕ → ℕ,
      StrictMono q ∧
      (∀ i, (q i).Prime) ∧
      (∀ i, q (i + 1) % q i = 1) ∧
      ∃ o : ℕ → ℝ,
        (o =o[atTop] (1 : ℕ → ℝ)) ∧
        -- We use `(k + 1)` here as the informal statement is 1-indexed.
        ∀ k, q k ≤ exp ((k + 1) * log (k + 1) ^ (1 + o k)) := by
…