O problema dos números congruentes: uma jornada da Antiguidade às curvas elípticas

Abstract

This dissertation focuses on the investigation of the problem of congruent numbers, a classic theme in Number Theory that asks which integers can be the area of a right triangle with rational sides. Although its formulation is elementary, the problem reveals remarkable depth, connecting historical mathematics to the frontiers of contemporary research, such as the Birch and Swinnerton-Dyer Conjectures. The work explores three equivalent formulations of the problem: the classical definition via areas of triangles, the perspective of arithmetic progressions of three rational squares, and the elegant connection with the theory of elliptic curves. Fundamental results are discussed, such as the noncongruence of the number 1 by Fermat and Tunnell's criterion, which represents the most significant advance toward a complete solution. As its main pedagogical contribution, this study presents, in an appendix, an educational product entitled "Exploring the World of Congruent Numbers in High School," designed to assist high school teachers in introducing this fascinating topic in the classroom.