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Level A · Machine-checkable Hard Number theory P-erdos-263

Erdős Problem #263

Is a_n = 2^2^n an irrationality sequence in the above sense?

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.

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Cite 
@misc{cairn-erdos-263,
  title        = {Erdős Problem #263},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-263}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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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

erdos_263.parts.i. Is an irrationality sequence in the above sense?

erdos_263.parts.ii. Must every irrationality sequence in the above sense satisfy as ?

Note: this was answered false for the pre-correction statement, which did not require monotonicity — the counterexample sequence is not increasing. The problem was corrected on erdosproblems.com on 2026-04-02 to require increasing sequences; for the corrected statement this question is open. The earlier formal proof (for the pre-correction definition) is preserved at https://github.com/google-deepmind/formal-conjectures/blob/c8cf651906abe91051cf835d4232ad5648412113/FormalConjectures/ErdosProblems/263.lean#L298

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«263» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_263.parts.i : answer(sorry) ↔ IsIrrationalitySequence (fun n : ℕ => 2 ^ 2 ^ n)
theorem erdos_263.parts.ii : answer(sorry) ↔
    ∀ a : ℕ → ℕ,
      IsIrrationalitySequence a →
        atTop.Tendsto (fun n : ℕ => (a n : ℝ) ^ (1 / (n : ℝ))) atTop

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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 — check erdosproblems.com/263. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/263

Source and licence

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