Aplicações da lei de reciprocidade quadrática e equações diofantinas não lineares no contexto olímpico.

Abstract

Given the growing expansion of mathematics olympiads in the country, preparing stu dents for contents that transcend the traditional curriculum constitutes a relevant challenge. In this context, the present work presents a detailed approach to the Law of Quadratic Reciprocity (LQR) and Non-Linear Diophantine Equations, using (Pólya, 1945) problem-solving perspective to highlight the importance of expanding students’ strategic arsenal when facing complex questions. Based on the selection and resolution of olympiad-level problems, this study seeks to establish a direct bridge between these advanced mathematical concepts and their practical application, demonstrating that mastering robust tools enriches comprehension and enables the simplification of intri cate problems. As an educational product resulting from this research, a self-sufficient instructional booklet was developed, compiling a miscellany of olympiad questions and illustrative examples related to the covered topics. This material aims to provide in depth theoretical and practical support, serving as a tool for both teaching practice and student preparation.