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WikipediaNumber theory

Schanuel's Conjecture

Given any set of nn complex numbers {z1,...,zn}\{z_1, ..., z_n\} that are linearly independent over Q\mathbb{Q}, the field extension Q(z1,...,zn,ez1,...,ezn)\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n}) has transcendence degree at least nn over Q\mathbb{Q}.

Number theoryAMS 11AMS 33

Mathematical statement

Given any set of nn complex numbers {z1,...,zn}\{z_1, ..., z_n\} that are linearly independent over Q\mathbb{Q}, the field extension Q(z1,...,zn,ez1,...,ezn)\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n}) has transcendence degree at least nn over Q\mathbb{Q}.

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

schanuel_conjecture

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem schanuel_conjecture (n : ℕ) (z : Fin n → ℂ) (h : LinearIndependent ℚ z) :    n ≤ Algebra.trdeg ℚ (adjoin ℚ (Set.range z ∪ Set.range (cexp ∘ z))) := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationWikipediabackground