FormalConjectures/ErdosProblems/285.lean

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L4DEL
Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 285

Reference: erdosproblems.com/285

Imports (1)
  1. theorem Erdos285.erdos_285 Documented declaration: erdos_285 sorry line 49 source

    Source documentation

    Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \dots < n_k$ with $$ 1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}. $$ Is it true that $$ f(k) = (1 + o(1)) \frac{e}{e - 1} k ? $$

    Proved by Martin [Ma00].

    [Ma00] Martin, Greg, Denser Egyptian fractions. Acta Arith. (2000), 231-260.

    depends on sorry what this means

    Statement uses: Iff, True, Nat, Set, Eq, Set.ofPred, Exists, Fin, Nat.succ, And, StrictMono, Not, Membership.mem, Set.range, OfNat.ofNat, Real, Finset.sum, Finset.univ, HDiv.hDiv, IsLeast and 7 more

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

    46 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_285 :
    answer(True) ↔ ∀ᵉ (f : ℕ → ℕ)
    (S : Set ℕ)
    (hS : S = {k | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧
      1 = ∑ i, (1 : ℝ) / n i })
    (h : ∀ k ∈ S,
      IsLeast
        { n (Fin.last k) | (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n)
          (_ : 1 = ∑ i, (1 : ℝ) / n i) }
        (f k)),
    ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)),
      ∀ k ∈ S, f k = (1 + o k) * rexp 1 / (rexp 1 - 1) * (k + 1) := by
…
  2. theorem Erdos285.erdos_285.variants.lb It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$ sorry line 67 source

    Source documentation

    It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$.

    depends on sorry what this means

    Statement uses: Nat, Set, Eq, Set.ofPred, Exists, Fin, Nat.succ, And, StrictMono, Not, Membership.mem, Set.range, OfNat.ofNat, Real, Finset.sum, Finset.univ, HDiv.hDiv, IsLeast, Fin.last, Asymptotics.IsLittleO and 6 more

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

    48 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_285.variants.lb (f : ℕ → ℕ)
    (S : Set ℕ)
    (hS : S = {k | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧
      1 = ∑ i, (1 : ℝ) / n i })
    (h : ∀ k ∈ S,
      IsLeast
        { n (Fin.last k) | (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n)
          (_ : 1 = ∑ i, (1 : ℝ) / n i) }
        (f k)) :
    ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)),
      ∀ k ∈ S, (1 + o k) * rexp 1 / (rexp 1 - 1) * (k + 1) ≤ f k := by
…