integer_complexity_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Integer complexity: ||n|| = minimum number of 1s to represent n using +, ×. Satisfies 3log₃(n) ≤ ||n|| ≤ 3log₂(n). The precise asymptotics and exact values for specific n are open.

Why this node was retired

The posted statement is

import Mathlib

noncomputable def intComplexity : ℕ → ℕ
    | 0 => 0
    | 1 => 1
    | n + 2 => (Finset.range (n + 2)).inf' ⟨0, by simp⟩
        (fun k => if h : 0 < k ∧ k < n + 2 then
          intComplexity k + intComplexity (n + 2 - k) else n + 2)

theorem integer_complexity_conjecture :
    ∀ n : ℕ, 1 ≤ n →
    3 * Nat.log 3 n ≤ intComplexity n ∧
    intComplexity n ≤ 3 * Nat.log 2 n + 3 := by
  sorry

The recursion permits addition only, omitting multiplication from integer complexity.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Define complexity as the minimum number of ones in expressions built using both addition and multiplication, or include multiplicative factor splits in the recursion. Reconcile floor-log constants with a stated theorem for that corrected definition.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.