Algebraic Numbers of the form αT with α Algebraic and T Transcendental

Štěpán Hubálovský,Eva Trojovská

doi:10.3390/math8101687

Abstract

Let α≠1 be a positive real number and let P(x) be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form αP(T), with T transcendental, is dense in some open interval of R.

Highlights

In 1900, in the International Congress of Mathematicians, Hilbert provided a list of 23 problems and his seventh problem was about the arithmetic nature of the power α β of two algebraic numbers α and β
The Gelfond–Schneider theorem classifies the nature of x y, for algebraic numbers x and y
Mathematics 2020, 8, 1687 the set of algebraic numbers of the form T T is dense in the interval [e−1/e, ∞)

Summary

Introduction

In 1900, in the International Congress of Mathematicians (in Paris), Hilbert provided a list of 23 problems and his seventh problem was about the arithmetic nature of the power α β of two algebraic numbers α and β. Mathematics 2020, 8, 1687 the set of algebraic numbers of the form T T (where T runs over all transcendental numbers) is dense in the interval [e−1/e , ∞). The set of algebraic numbers of the form α P(T ) , with T transcendental, is dense in some open interval of R. For α = 2 and P( x ) = x, we have that the previous result implies in the existence of an open interval I ⊆ R such that the set of algebraic numbers of the form 2T with T transcendental, is dense in I. We remark that Schanuel’s conjecture is proved only for n = 1: Lindemann’s theorem asserts that is a transcendental number, for any non-zero algebraic number α. The number e can be replaced by some other known transcendental numbers, as π, log 2, etc

Proof of the Theorem 1

Proof of the Theorem 2

Conclusions