Schanuel's conjecture

Remarks

The conjecture is open. The case $n = 1$ is known: if $z \ne 0$, then $z$ or $e^z$ is transcendental, which is the Hermite–Lindemann theorem. If $z_1, \dots, z_n$ are algebraic, the conjecture gives the Lindemann–Weierstrass theorem, which is proved. With $z_1 = 1$ and $z_2 = i\pi$, it would imply that $e$ and $\pi$ are algebraically independent; it is not even known whether $e + \pi$ is irrational. It would also imply the Gelfond–Schneider theorem and Baker's theorem, and even that $\Q$-linearly independent logarithms of algebraic numbers are algebraically independent.