FormalConjectures/ErdosProblems/258.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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 258

References:

Imports (1)
  1. theorem Erdos258.erdos_258 sorry line 42 source

    depends on sorry what this means

    Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Tendsto, Filter.atTop, Irrational, tsum, Real, HDiv.hDiv, Finset.card, Nat.divisors, HAdd.hAdd, Finset.prod, Finset.Icc, SummationFilter.unconditional

    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.

  2. theorem Erdos258.erdos_258.variants.monotone Let $2 \leq a_1 \leq a_2 \leq \cdots$ be a monotone sequence with $a_n \to \infty$ sorry line 55 source

    Source documentation

    Let $2 \leq a_1 \leq a_2 \leq \cdots$ be a monotone sequence with $a_n \to \infty$. Is $\sum_n \frac{d(n)}{a_1 \cdots a_n}$ irrational, where $d(n)$ is the number of divisors of $n$?

    Solution: True (proved by Erdős and Straus [ErSt71], Lemma 2.2 and Theorem 2.13).

    depends on sorry what this means

    Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Monotone, Filter.Tendsto, Filter.atTop, Irrational, tsum, Real, HDiv.hDiv, Finset.card, Nat.divisors, HAdd.hAdd, Finset.prod, Finset.Icc, SummationFilter.unconditional

    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_258.variants.monotone : answer(True) ↔
    ∀ (a : ℕ → ℕ), (∀ n, 2 ≤ a n) → Monotone a →
    Filter.Tendsto a Filter.atTop Filter.atTop →
    Irrational (∑' (n : ℕ), ((n + 1).divisors.card / ∏ i ∈ Finset.Icc 1 (n + 1), a i)) := by
…
  3. theorem Erdos258.erdos_258.variants.constant sorry line 69 source

    depends on sorry what this means

    Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Irrational, tsum, Real, HDiv.hDiv, Finset.card, Nat.divisors, HAdd.hAdd, HPow.hPow, SummationFilter.unconditional

    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.