Rationality and power

2014/09/12 by Sam Chow, Bin Wei, Chow, Sam +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #History and Theory of Mathematics #Mathematical and Theoretical Analysis #math.NT #msc:11A99 #msc:11D61 #msc:11J72 #msc:11J81

paper · pdf · doi:10.48550/arxiv.1409.3790

arxiv created 2014/09/12 · arxiv updated 2014/09/15

Abstract

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation x^x = α, where α is a fixed algebraic number. We then explore the consequences of x^P(x) being rational, if x is rational and P(x) is a fixed integer polynomial.

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