Banca de QUALIFICAÇÃO: MARIA ELOISA FERREIRA DOS SANTOS

Uma banca de QUALIFICAÇÃO de MESTRADO foi cadastrada pelo programa.
STUDENT : MARIA ELOISA FERREIRA DOS SANTOS
DATE: 16/11/2023
TIME: 09:00
LOCAL: Campus Arapiraca
TITLE:

Transcendent numbers and Equations in form x^n=n^x

KEY WORDS:
PAGES: 41
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

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 the 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 $n \in \mathbb{N}$ 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 $x^n=n^x$, with $n \in \mathbb{N}-\{0,1\}$ and $x \in \mathbb{ R}-\{0,1\}$ 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. Finally, two didactic sequences will be presented as an incentive to approach transcendent numbers in high school and teacher training.

COMMITTEE MEMBERS:
Presidente - 1422462 - ALCINDO TELES GALVAO
Interno(a) - 1824484 - MORENO PEREIRA BONUTTI