Math Conjectures

Schanuel’s Conjecture

OPENLandmarkConjectureProposed c. 1960 · Full conjecture

Canonical statement

For every integer n≥1n\ge1, if z1,…,zn∈Cz_1,\ldots,z_n\in\mathbb C are linearly independent over Q\mathbb Q, then
trdeg⁡QQ(z1,…,zn,ez1,…,ezn)≥n, \operatorname{trdeg}_{\mathbb Q} \mathbb Q(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})\ge n,
where trdeg⁡Q\operatorname{trdeg}_{\mathbb Q} denotes transcendence degree over Q\mathbb Q.
View source LaTeX
For every integer \(n\ge1\), if \(z_1,\ldots,z_n\in\mathbb C\) are linearly independent over \(\mathbb Q\), then
\[
  \operatorname{trdeg}_{\mathbb Q}
    \mathbb Q(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})\ge n,
\]
where \(\operatorname{trdeg}_{\mathbb Q}\) denotes transcendence degree over \(\mathbb Q\).
NT-TRANS-021:[Lang1966Transcendental][MacintyreWilkie1996][Marker2006Schanuel]
Number theoryLogic and foundations

Notes

Schanuel's conjecture asserts that if z1,…,znz_1,\ldots,z_n are complex numbers linearly independent over Q\mathbb Q, then the field Q(z1,…,zn,ez1,…,ezn)\mathbb Q(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n}) has transcendence degree at least nn over Q\mathbb Q. Schanuel circulated the conjecture orally around 1960, and an early printed source is Lang's 1966 book on transcendental numbers [Lang1966Transcendental], so the proposal date is approximate.

The case n=1n=1 follows from the Lindemann–Weierstrass theorem, but already n=2n=2 outstrips known methods: taking z1=1z_1=1 and z2=iπz_2=i\pi, the conjecture would give the algebraic independence of ee and π\pi, which is unproven — it is not even known that e+πe+\pi is irrational. The conjecture would subsume the classical transcendence theory of the exponential function. It also has a life in logic: Macintyre and Wilkie showed that decidability of the first-order theory of the real exponential field follows from a real form of Schanuel's conjecture [MacintyreWilkie1996], and the conjecture is central to Zilber's pseudoexponentiation program [Marker2006Schanuel].

It remains open; existing transcendence machinery does not control the joint transcendence degree of the 2n2n numbers involved.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.