SCHANUEL’S CONJECTURE AND ALGEBRAIC POWERS z w AND wz WITH z AND w TRANSCENDENTAL

Abstract

We give a brief history of transcendental number theory, including

Schanuel’s conjecture (S). Assuming (S), we prove that if z and w are

complex numbers, not 0 or 1, with z w and w z algebraic, then z and w

are either both rational or both transcendental. A corollary is that if

(S) is true, then we can find transcendental positive real numbers x, y,

and s = t such that the three numbers x y = y x and s t = t s are all

integers. Another application (possibly known) is that (S) implies the

transcendence of the numbers

√ 2

√2 √2

, i i i

, i e

π

.

We also prove that if (S) holds and α α

z

= z, where α = 0 is algebraic

and z is irrational, then z is transcendental.