SCHANUEL’S CONJECTURE AND ALGEBRAIC POWERS zw AND wz WITH z AND w TRANSCENDENTAL
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 xy = y x and st = ts 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 αα z = z, whereα = 0 is algebraic and z is irrational, then z is transcendental.