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Level A · Machine-checkable Hard Number theory P-mathoverflow-75792

Mathoverflow 75792

Is 2n the complexity of 2^n for 0 < n?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-mathoverflow-75792,
  title        = {Mathoverflow 75792},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mathoverflow-75792}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Is 2n the complexity of 2^n for 0 < n?

Various questions about integer complexity, which is the minimum number of 1s needed to express a natural number using addition, multiplication, and parentheses.

Let ‖n‖ denote the integer complexity of n > 0.

  • It is known that ‖3^n‖ = 3n for n > 0.
  • Is it true that ‖2^n‖ = 2n for n > 0?
  • The corresponding conjecture for 5 is false, because 5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)!

We have chosen to formalise this using an inductive type.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Mathoverflow.«75792». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem complexity_two_pow : answer(sorry) ↔ ∀ n : ℕ, 0 < n → complexity (2 ^ n) = 2 * n

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • mathoverflow/75792 by user Harry Altman
  • http://arxiv.org/abs/1203.6462 by Jānis Iraids, Kaspars Balodis, Juris Čerņenoks, Mārtiņš Opmanis, Rihards Opmanis, Kārlis Podnieks
  • http://arxiv.org/abs/1207.4841 by Harry Altman, Joshua Zelinsky
  • https://oeis.org/A5245 : Mahler-Popken complexity.

Source and licence

Imported from Formal Conjectures (MathOverflow), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.