Teoria dos números: uma análise sobre problemas em aberto

Abstract

This monograph work aims to present an analysis of the conjectures and open problems

in the study of Number Theory. We will cover basic concepts to understand this analy- sis, such as the concept of prime numbers, its infinity, its applications and its formation

pattern, which is the first elementary open problem analyzed in this work. Moreover, we will also see other open problems involving prime numbers, such as the famous Mersenne Primes and twin primes, showing the importance of the study of prime number theory.

In the field of conjectures, we will begin by studying one of the most beautiful conjec- tures presented: The Goldbach’s Conjecture. This conjecture has a relatively simple

statement to understand and an extremely relevant importance for Number Theory. We

will also present the Legendre Conjecture, which resumes the applications of prime num- bers, showing again how mysterious these numbers are and their extreme importance for

Mathematics. We will also see the Collatz Conjecture, an important statement for prime number theory and full of applications, especially in the field of Computing. We will also contemplate other open problems, in which we will see the so-called perfect numbers and friendly numbers. The study of these numbers is extremely important for Number Theory, since it recovers basic concepts such as divisibility and modular congruence.