FormalConjectures/ErdosProblems/673.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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 673

References:

Imports (1)
  1. def Erdos673.G If $1=d_1<\cdots <d_{\tau(n)}=n$ are the divisors of $n$ (here Nat.nth (· ∣ n) i is $d_{i+1}$), then $$G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.$$ line 39 source

    Source documentation

    If $1=d_1<\cdots <d_{\tau(n)}=n$ are the divisors of $n$ (here Nat.nth (· ∣ n) i is $d_{i+1}$), then $$G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.$$

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

    Statement uses: Nat, Real

    Proof uses: Finset.sum, Fin, HSub.hSub, Finset.card, Nat.divisors, OfNat.ofNat, Finset.univ, HDiv.hDiv, Nat.nth, Dvd.dvd, HAdd.hAdd

    14 instances, coercions and auxiliary constants left out.

    noncomputable def G (n : ℕ) : ℝ :=
…
  2. theorem Erdos673.erdos_673.parts.i sorry line 53 source

    depends on sorry what this means

    Statement uses: Iff, True, Real, Set.HasDensity, Nat, Set.ofPred, LT.lt, Erdos673.G, OfNat.ofNat, Set.univ

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

    15 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem Erdos673.erdos_673.parts.ii sorry line 63 source

    depends on sorry what this means

    Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Finset.sum, Finset.Icc, OfNat.ofNat, Erdos673.G, HMul.hMul, Real.log

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

    26 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  4. theorem Erdos673.erdos_673.variants.average sorry line 70 source

    depends on sorry what this means

    Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Finset.sum, Finset.Icc, OfNat.ofNat, Erdos673.G, Filter.atTop

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

    20 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  5. theorem Erdos673.erdos_673.variants.tao Tao observed that, for any divisor $m\mid n$ with $m>1$, $\frac{\tau(n/m)}{m} \leq G(n) \leq \tau(n)$ sorry line 77 source

    Source documentation

    Tao observed that, for any divisor $m\mid n$ with $m>1$, $\frac{\tau(n/m)}{m} \leq G(n) \leq \tau(n)$.

    depends on sorry what this means

    Statement uses: Nat, LT.lt, OfNat.ofNat, Dvd.dvd, And, LE.le, Real, HDiv.hDiv, Finset.card, Nat.divisors, Erdos673.G

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

    20 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_673.variants.tao (n m : ℕ) (hn : 0 < n) (hm : 1 < m) (hmn : m ∣ n) :
    ((n / m).divisors.card : ℝ) / m ≤ G n ∧ G n ≤ n.divisors.card := by
…

    Mentions: Erdos673.G