Quadrados perfeitos e números p-ádicos: aplicações ao ensino e à questão 6 da IMO de 1988

Abstract

This dissertation proposes a method for solving mathematical olympiad problems involving perfect squares, aiming to overcome the inherent limitation of the modular contradiction method, which only verifies the non-existence of solutions. We developed an approach based on the local–global principle, using p-adic numbers and the Grunwald–Wang theorem to establish conditions that guarantee when an integer is a perfect square. The method is applied to the historic Problem 6 of the 1988 IMO, offering an alternative solution to approaches such as Vieta Jumping and Fermat’s Infinite Descent. Beyond the theoretical contributions, the work includes didactic proposals for basic education, with investigative activities on 2-adic numbers and Pythagorean dates, demonstrating how advanced mathematical concepts can be made accessible to basic education students. The results indicate the feasibility of integrating number theory with mathematics education, pointing toward applications in other olympiad problems.