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Level A · Machine-checkable Hard Number theory P-mathoverflow-17560

Mathoverflow 17560

If 2^x and 3^x are integers, then x must be an integer.

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-17560,
  title        = {Mathoverflow 17560},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mathoverflow-17560}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Disputed
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Refuted
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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

If and are integers, then must be an integer.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Mathoverflow.«17560».

theorem mathoverflow_17560 {x : ℝ} (hx : ∃ m : ℕ, (2 : ℝ) ^ x = m) (hx' : ∃ m : ℕ, (3 : ℝ) ^ x = m) :
    ∃ m : ℕ, x = m

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/17560 asked by user Alon-Amit

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.