Representação de inteiros por formas quadráticas binárias.

Abstract

This work consists in presenting answers to the following questions: be the quadratic form ax2 + bxy + cy2, discriminant Δ = b2-4ac, in two variables x and y, with a, b, c integers given, not all 0; for which integers n are integers x and y such that n = ax2 + bxy + cy2? What is the characterization of positive integers that can be written as the sum of two squares? What are the primes p> 3 that can be represented by the form 2x2 + 3y2 or by the form x2 + 6y2? In this dissertation, we study the mathematical theories that allow the resolution of the questions discussed above. In this sense, we present a good part of the prerequisites essential to the appreciation of the central results that we have discussed. Then, we discuss the representation of integers by binary quadratic forms, establishing a useful criterion, from which we can determine if an integer n is or not representable by some quadratic form, given its discriminant. Finally, we present answers to the last two questions and we weave our considerations regarding the work produced, pointing to what goes beyond the theory developed here and recognizing the relevance of the interrelations between different areas of Mathematics.