Inteiros que são soma de dois quadrados.

Abstract

The main objective of this monograph is to characterize the integers that can be written as the sum of two squares. This is a classical result in Number Theory, whose discovery is attributed to Fermat (1601—1665). The essential part of the proof consists in characterizing the prime numbers that are sums of two squares; more precisely, if 𝑝 > 2 is a prime and can be expressed as the sum of two squares, then 𝑝 is of the form 4𝑘 + 1. This statement was sent by Fermat to Mersenne (1588—1648) in a letter dated 1640. The mathematician Euler (1707—1783) provided a proof of this result in 1754. In this monograph, we present a modern proof of this result, following the classical approach based on properties of the ring of Gaussian integers, which are complex numbers of the form 𝑧 = 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are integers and 𝑖 2 = −1 .