Congruências: resíduos quadráticos e representação de inteiros como soma de quadrados

Abstract

Number Theory is one of the major areas of mathematics and the focus of studies in this area is the behavior of numbers (mainly integer) under specific conditions. Some relationships can be made between these numbers or structures that are studied, such

as the divisibility. This relationship was explored by many mathematicians and its ad- vancement resulted in congruences. The congruences, especially the congruences module

m, with m > 0 and integer (which are explored in this research), bring many interesting results such as the concept of residue. It is known that the any division leaves a residue, and is no different with a number squared, which leave the so-called Quadratic Residues. And it was through this important definition, combined with the definition of Legendre Symbol, that it was possible to understand how the Representation of Integers as Sum of Squares takes place. Due to these mentioned aspects, the present work consists of a bibliographic review that aims to study this theme. To this end, the works were analyzed: Hefez (2016), Burton (2016), Santos (2009), Merzbach (2012) and O’Regan (2016), which present a good organization from the Congruences to the Representation of Integers as the Sum of Squares. The conclusion of the work is that every natural number can be expressed as the sum of a maximum of 4 squares (result proved by the mathematician Lagrange), and to achieve this result, important tools such as Fermat’s Little Theorem, Euler’s Theorem and Wilson’s Theorem were used, in addition to several other previous

results presented, especially in Congruences and Quadratic Residues. Finally, in the Ap- pendix, we present a table with the “smallest representation” as the sum of squares for the first 100 naturals and we included a theorem enunciated and demonstrated by the mathematician Legendre that talks about numbers that can only be expressed as the sum of four squares.