The authors prove several conditional results on transcendence assuming Schanuel’s conjecture. The conjecture states that if numbers are linearly independent over then there are at least algebraically independent numbers among .
Under this conjecture the authors prove that if for the numbers and are algebraic, then and are either both rational or both transcendental. This, for instance, implies the transcedence of the numbers and .
Another conditional result is that if and are complex numbers with algebraic and irrational such that , then is transcendental.