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Level A · Machine-checkable Hard Analysis P-irrational

Open questions on irrationality of numbers

Are e and π algebraically independent?

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-irrational,
  title        = {Open questions on irrationality of numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/irrational}},
  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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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

algebraicIndependent_e_pi. Are and algebraically independent?

irrational_e_plus_pi. Is irrational?

irrational_e_times_pi. Is irrational?

irrational_e_to_e. Is irrational?

irrational_pi_to_e. Is irrational?

irrational_pi_to_pi. Is irrational?

irrational_ln_pi. Is irrational?

irrational_eulerMascheroniConstant. Is the Euler-Mascheroni constant irrational?

irrational_catalanConstant. Is the Catalan constant irrational?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Irrational (9 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem algebraicIndependent_e_pi :
    answer(sorry) ↔ AlgebraicIndependent ℚ ![e, π]
theorem irrational_e_plus_pi :
    answer(sorry) ↔ Irrational (e + π)
theorem irrational_e_times_pi :
    answer(sorry) ↔ Irrational (e * π)
theorem irrational_e_to_e :
    answer(sorry) ↔ Irrational (e ^ e)
theorem irrational_pi_to_e :
    answer(sorry) ↔ Irrational (π ^ e)
theorem irrational_pi_to_pi :
    answer(sorry) ↔ Irrational (π ^ π)
theorem irrational_ln_pi :
    answer(sorry) ↔ Irrational (log π)
theorem irrational_eulerMascheroniConstant :
    answer(sorry) ↔ Irrational eulerMascheroniConstant
theorem irrational_catalanConstant :
    answer(sorry) ↔ Irrational catalanConstant

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. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

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.