NÚMEROS TRANSCENDENTES E AS EQUAÇÕES DA FORMA X^N = N^X

Abstract

Of the many unresolved problems in Mathematics, some are concepts and elements arising from the Theory of  Transcendent Numbers, for example the difficulty in demonstrating that the nature of a number is transcendental. Based on advances in this theory, one of the results that is extremely important for "constructing" a transcendent number in the form of a power is the Gelfond-Schneider Theorem. Inserted in this scenario of transcendent powers, the nature of powers of the form n^T, with n \in N and T transcendent, is little known. Regarding the numbers 2^\pi and 2^e, for example, it is not yet known whether they are transcendent or not. Therefore, in this work we carried out a study on the solutions of the equation xⁿ=n^x, with n \in N-{0,1} and x \in R-{0,1} and its relationship with transcendent numbers of the form n^T, within the conditions presented. With this, we define a transcendence criterion for such powers and also highlight that such a result is not unique, there are other transcendent numbers that do not meet this criterion, as well as there are numbers of the form n^T that are algebraic.

Keywords: Algebraic numbers; transcendent powers; Gelfond-Schneider theorem.