Exponentials conjectures and theorems

Four exponentials conjecture Let x_0, x_1 and y_0, y_1 be ℚ-linearly independent pairs of complex numbers, then some e^x_i y_j is transcendental.

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

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

four_exponentials_conjecture. Four exponentials conjecture Let $x_{0}, x_{1}$ and $y_{0}, y_{1}$ be $Q$ -linearly independent pairs of complex numbers, then some $e^{x_{i} y_{j}}$ is transcendental.

two_pow_three_pow_transcendental. The four exponential conjecture would imply that for any irrational number $t$ , at least one of the numbers $2^{t}$ and $3^{t}$ is transcendental.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Exponentials (2 statements).

theorem four_exponentials_conjecture (x : Fin 2 → ℂ) (y : Fin 2 → ℂ)
    (h1 : LinearIndependent ℚ x) (h2 : LinearIndependent ℚ y) :
    ∃ i j : Fin 2, Transcendental ℚ (exp (x i * y j))

theorem two_pow_three_pow_transcendental (t : ℝ) (h : Irrational t) :
    Transcendental ℚ (2 ^ t : ℝ) ∨ Transcendental ℚ (3 ^ t : ℝ)

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.