OP-00013 Open

The four exponentials conjecture

Let $x_1, x_2$ be complex numbers that are linearly independent over $\mathbb{Q}$, and let $y_1, y_2$ be as well. Then at least one of the four numbers $e^{x_i y_j}$, $i, j \in \{1, 2\}$, is transcendental.

Source: Th. Schneider, Einführung in die transzendenten Zahlen (1957); stated explicitly by Lang (1966) and Ramachandra, Acta Arith. 14 (1968).

Classification

Primary: NT — Number Theory
Keywords: exponential function, transcendental numbers

References

@book{schneider1957,
  author = {Schneider, Theodor},
  title = {Einf{\"u}hrung in die transzendenten Zahlen},
  publisher = {Springer},
  year = {1957}
}

@book{lang1966,
  author = {Lang, Serge},
  title = {Introduction to Transcendental Numbers},
  publisher = {Addison-Wesley},
  year = {1966}
}

@article{ramachandra1968,
  author = {Ramachandra, K.},
  title = {Contributions to the theory of transcendental numbers},
  journal = {Acta Arithmetica},
  volume = {14},
  year = {1968}
}

@misc{wikipedia,
  title = {Four exponentials conjecture},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Four_exponentials_conjecture}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/Exponentials.lean#L32},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00013.

/-- OP-00013: The four exponentials conjecture -/
def Problem_OP_00013 : Prop :=
  ∀ (x y : Fin 2 → ℂ),
      LinearIndependent ℚ x → LinearIndependent ℚ y → ∃ i j, Transcendental ℚ (Complex.exp (x i * y j))

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00013

theorem resolution : RegistryProblems.Problem_OP_00013 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00013"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00013 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00013, "The four exponentials conjecture", clopen.solutions.