[#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.

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1Context

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

2Definitions

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.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Four exponentials conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details 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. Open full record →
Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic.^[1] Relevance to this problem For Four exponentials conjecture, record four-exponentials-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove the transcendence conclusion for every pair of rationally independent pairs, or give qualifying pairs for which all four exponentials are algebraic. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Four exponentials conjecture.” TheoremDB. P4. Problem statement; statement text SHA-256 ea474ced7e54853c5c0b3c647772b88012fa0ac94266523e2217238392cc1341. https://theoremdb.org/statement/?ref=P4

BibTeX

@misc{theoremdb-problem-ea474ced7e54853c5c0b3c647772b88012fa0ac94266523e2217238392cc1341,
  title = {{Four exponentials conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 ea474ced7e54853c5c0b3c647772b88012fa0ac94266523e2217238392cc1341},
  url = {https://theoremdb.org/statement/?ref=P4}
}

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This problem includes 2 records joined by 2 typed links, sourced from doi.org^[1], current as of July 31, 2026.

1Lean verification

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.

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[#P4] Four exponentials conjecture

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