Open questions on irrationality of numbers

Are e and π algebraically independent?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	8 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

algebraicIndependent_e_pi. Are $e$ and $π$ algebraically independent?

irrational_e_plus_pi. Is $e + π$ irrational?

irrational_e_times_pi. Is $e π$ irrational?

irrational_e_to_e. Is $e^{e}$ irrational?

irrational_pi_to_e. Is $π^{e}$ irrational?

irrational_pi_to_pi. Is $π^{π}$ irrational?

irrational_ln_pi. Is $ln (π)$ irrational?

irrational_eulerMascheroniConstant. Is the Euler-Mascheroni constant $γ$ irrational?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Irrational (8 statements).

theorem algebraicIndependent_e_pi :
    AlgebraicIndependent ℚ ![e, π]

theorem irrational_e_plus_pi :
    Irrational (e + π)

theorem irrational_e_times_pi :
    Irrational (e * π)

theorem irrational_e_to_e :
    Irrational (e ^ e)

theorem irrational_pi_to_e :
    Irrational (π ^ e)

theorem irrational_pi_to_pi :
    Irrational (π ^ π)

theorem irrational_ln_pi :
    Irrational (log π)

theorem irrational_eulerMascheroniConstant :
    Irrational eulerMascheroniConstant

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
[OAI26a] OpenAI, Catalan's constant is irrational. OpenAI Math Release preprint (2026). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.