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Level A · Machine-checkable Hard Number theory P-book-bugeaud-distribution-modulo-one-problem10-9

Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers

Problem 10.9. There are no real numbers ξ such that 0 ≤ ξ (3/2)^n < 1/2 for every positive integer n, i.e. no Z-number exists. Posed by Mahler [Mah68].

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-book-bugeaud-distribution-modulo-one-problem10-9,
  title        = {Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/book-bugeaud-distribution-modulo-one-problem10-9}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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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

Problem 10.9. There are no real numbers such that for every positive integer , i.e. no Z-number exists. Posed by Mahler [Mah68].

See also FormalConjectures/Wikipedia/Mahler32.lean.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9.

theorem problem_10_9 : type_of% Mahler32.mahler_conjecture

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

  • [Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.
  • [Mah68] Mahler, Kurt. "An unsolved problem on the powers of 3/2." Journal of the Australian Mathematical Society 8.2 (1968): 313-321.
  • [FLP95] Flatto, Leopold, Jeffrey C. Lagarias, and Andrew D. Pollington. "On the range of fractional parts ." Acta Arithmetica 70.2 (1995): 125-147.

Source and licence

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