FormalConjectures/ErdosProblems/243.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 243

Reference: erdosproblems.com/243

Imports (1)
  1. theorem Erdos243.erdos_243 Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$ sorry line 41 source

    Source documentation

    Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$.

    Then, for all sufficiently large $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.

    depends on sorry what this means

    Statement uses: Nat, StrictMono, Filter.Tendsto, Real, HDiv.hDiv, HPow.hPow, HSub.hSub, OfNat.ofNat, Filter.atTop, nhds, Summable, Rat, SummationFilter.unconditional, Filter.Eventually, Eq, HAdd.hAdd

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

    60 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_243 (a : ℕ → ℕ) (ha₀ : StrictMono a)
    (ha₁ : Tendsto (fun n ↦ (a n : ℝ) / a (n - 1) ^ 2) atTop (𝓝 1))
    (ha₂ : Summable ((1 : ℚ) / a ·)) :
      ∀ᶠ n in atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1 := by
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