Números primos: fundamentos, propriedades e potencial didático

Abstract

This dissertation provides a comprehensive study of prime numbers, combining their theoretical and historical foundations with pedagogical practice in elementary education. The central objective of the work is to develop a teaching approach that goes beyond memorization, investigating the topic's potential to promote more contextualized learning. The research begins with the history of number theory, tracing the evolution of the concept of primes from the Pythagoreans and Euclid's rigorous formalization, to its revival in the 17th century through the works of Fermat and Mersenne. The mathematical foundation was built in chapters dedicated to the foundations of divisibility. The study delves into essential properties, such as the Fundamental Theorem of Arithmetic (FTA), which guarantees the unique decomposition of a number into prime factors, the classic proof of the infinity of primes, the sieve of Eratosthenes method, and Fermat's little theorem. To illustrate the ongoing research in the field, famous and still unresolved conjectures, such as Goldbach's and the twin prime conjecture, are presented. The dissertation focuses on the Brazilian educational landscape, analyzing the topic's presence in the National Common Curricular Base (BNCC) and discussing the recurring difficulties in teaching and learning. An analysis of textbooks reveals that, although they follow a logical structure, they often miss the opportunity to connect the content to its modern applications and intrinsic beauty. As its main contribution, the work presents an educational product, a detailed teaching sequence with five lessons. The proposal uses investigative strategies, such as the construction of the Sieve of Eratosthenes, the exploration of the proof of the infinity of primes, and the discussion of conjectures, seeking to spark curiosity and develop critical thinking in students.