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Algebraic and transcendental solutions of some exponential equations. (English) Zbl 1224.11059

Summary: We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations \(x^x = y\), \(x^y = y^x\), \(x^x = y^y\), \(x^y = y\), and \(x^{x^y}= y\). Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.

MSC:

11D61 Exponential Diophantine equations
11J85 Algebraic independence; Gel’fond’s method
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