[#P4] Four exponentials conjecture
Problem. Let \(x_1,x_2\in\mathbb{C}\) be linearly independent over \(\mathbb{Q}\), and let \(y_1,y_2\in\mathbb{C}\) be linearly independent over \(\mathbb{Q}\). At least one of the four numbers \(e^{x_i y_j}\), with \(i,j\in\{1,2\}\), is transcendental.
1Context
The conjecture asks whether the known six exponentials theorem can be sharpened to the first unresolved two-by-two case.
2Problem setup
Definition 1 (Linear independence over Q). Linear independence over Q means that no nonzero rational linear combination of the two numbers is zero.
Definition 2 (A complex number). A complex number is transcendental when it is not a root of any nonzero polynomial with rational coefficients.
Remark 1. The conjecture asks whether the known six exponentials theorem can be sharpened to the first unresolved two-by-two case.
3What counts as a solution
- Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The six exponentials theorem proves the analogous 2 by 3 statement, and Schanuel's conjecture implies the four exponentials target. Waldschmidt's survey chapter records the complex four-exponentials problem as open. Exact unresolved remainder: For every two Q-linearly independent x-values and two Q-linearly independent y-values, prove that at least one of the four exponentials is transcendental, or give qualifying pairs for which all four are algebraic.[2][1]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
Original intake status. The cited survey chapter identifies the four exponentials problem as open in the complex case. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- The six exponentials theorem is known. Schanuel's conjecture would imply this four-number assertion.
2See also
How to cite
TheoremDB contributors, “Four exponentials conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/four-exponentials-conjectureThis page as plain text: four-exponentials-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from doi.org[1], current as of July 31, 2026.
1References
- Packet source. Michel Waldschmidt, “The Four Exponentials Problem and Schanuel’s Conjecture”. Lecture Notes in Mathematics (2023), 579-592. DOI 10.1007/978-3-031-12244-6_39. Michel Waldschmidt, Mathematics Going Forward, 2022, chapter 39. ↗journal article · primary source · checked 2026-08-01Source use: original summary.The cited survey chapter identifies the four exponentials problem as open in the complex case. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at chapter 39, formulation and comparison with six exponentials and Schanuel.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Gives the theorem-conjecture boundary for the exact complex four-exponentials target.Source named by the research packet.
- K. Ramachandra, “Contributions to the theory of transcendental numbers (II)”. Acta Arithmetica 14(1) (1968), 73-88. DOI 10.4064/aa-14-1-73-88. section 4, especially the corollaries of Theorem 2 on pages 87-88. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the neighboring Six Exponentials Theorem; the four-exponential target removes the third independent y-value.
An original CC0 restatement prepared by TheoremDB maintainers.