Scholz conjecture on addition chains

The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer n, the addition-chain length of 2^n - 1 is at most n - 1 + ℓ(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-scholz-conjecture,
  title        = {Scholz conjecture on addition chains},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/scholz-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer $n$ , the addition-chain length of $2^{n} - 1$ is at most $n - 1 + ℓ (n)$ .

Formal statement (Lean 4)

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

theorem scholz_conjecture :
    answer(sorry) ↔ ∀ (n : ℕ), 0 < n → ℓ(2 ^ n - 1) ≤ n - 1 + ℓ(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

Wikipedia
MathWorld
Tall22 Amadou Tall. "The Scholz conjecture on addition chain is true for infinitely many integers with $ℓ (2 n) = ℓ (n)$ ." _arXiv:2210.13812_ (2022). Also available as ePrint 2023/020.
OEIS A003313

Source and licence

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

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