# P4: Four exponentials conjecture

- ID: `P4`
- Reference: `four-exponentials-conjecture`
- Page: https://theoremdb.org/statements/P4
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

The conjecture asks whether the known six exponentials theorem can be sharpened to the first unresolved two-by-two case.

### Problem setup

- **Definition (Linear independence over Q).** Linear independence over Q means that no nonzero rational linear combination of the two numbers is zero.
- **Definition (A complex number).** A complex number is transcendental when it is not a root of any nonzero polynomial with rational coefficients.
- **Remark.** The conjecture asks whether the known six exponentials theorem can be sharpened to the first unresolved two-by-two case.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against Waldschmidt's survey on 2026-07-22.
- The six exponentials theorem is known. Schanuel's conjecture would imply this four-number assertion.

### Open directions

- **Route 1** (reported): Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `four-exponentials-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.1007/978-3-031-12244-6_39
   - Also cited at chapter 39, formulation and comparison with six exponentials and Schanuel
   - Also cited at Editorial research route recorded 2026-07-31
   - journal_article; primary source; checked 2026-08-01
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.4064/aa-14-1-73-88
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves the neighboring Six Exponentials Theorem; the four-exponential target removes the third independent y-value.