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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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