Eventual existence of tail divisor adjacent chain list

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Erdos390.eventual_tail_divisor_chain_list_exists

by doctosil · Sep 6, 2026 · Mathlib c5ea003 (Lean v4.30.0)

asymptoticscombinatoricsnumber-theory

Fix a constant $c > C_0 = \frac{4029639598}{25970038185}$ and put $h = \left\lceil c \frac{n}{\log n} \right\rceil$ . For all sufficiently large $n \in \mathbb{N}$ , given any divisor $D \in \mathbb{N}$ and central subset $\mathrm{central} \subseteq (n, 2n]$ such that $\prod \mathrm{central} = \binom{2n}{n} D$ and $D \mid \prod_{(2n, 2n + h]} k$ , there exists an integer list $l$ in $(2n, 2n + h]$ satisfying the adjacent chain condition whose product equals $D$ :

l.\mathrm{IsChain} (<) \quad \text{and} \quad \prod_{x \in l} x = D.

Preamble

import Definitions.Def_erdos390_problem
open Filter
open Erdos390

Formal statement

namespace Erdos390

theorem eventual_tail_divisor_chain_list_exists :
    ∀ c : ℝ, C0 < c →
      ∀ᶠ n : ℕ in atTop,
        ∀ (D : ℕ) (central : Finset ℕ),
          central ⊆ factorInterval n (2 * n) →
          central.prod id = Nat.choose (2 * n) n * D →
          D ∣ (factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))).prod id →
          ∃ l : List ℕ,
            (∀ x ∈ l, x ∈ factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))) ∧
            l.IsChain (· < ·) ∧
            l.prod = D := by sorry

end Erdos390

Source

Shouqiao Wang, A Proposed Solution to Erdős Problem 390, Section 10, BankPaperGuardedUpperProductAssembly.lean

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