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.
Current state (tracked automatically)
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Formal Conjectures: research statements in the file | 2 open, 0 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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/exponentials)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.
- Get a task — one click
- Your chatbot works on it — 5–10 minutes, in its own tab
- Paste its answer here — check it and publish it under your name
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.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
four_exponentials_conjecture. Four exponentials conjecture Let and be -linearly independent pairs of complex numbers, then some is transcendental.
two_pow_three_pow_transcendental. The four exponential conjecture would imply that for any irrational number , at least one of the numbers and 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
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.