Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

Details Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

2010-10-29

arxiv.org/abs/1010.6216v3

East-West J. Math. 12, no. 1 (2010) 75-84

Mathematics

Number Theory

6 pages. Corrected a mistake in the paragraph after (4.5), updated [15]

Scientific paper

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 four distinct transcendental positive real numbers x, y, 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 sqrt(2)^sqrt(2)^sqrt(2), i^i^i, and i^e^pi. We also prove that if (S) holds and a^a^z=z, where a=/=0 is algebraic and z is irrational, then z is transcendental.

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