Seeing Arithmetic: A Matrix and Congruence Approach to the Twin Prime Conjecture in High School

This work originates within the framework of the Liceo Matematico, a national Italian project of didactic experimentation that supplements and enriches the traditional Scienti c High School. The curriculum o¤ers additional hours dedicated to mathematical deepening and interdisciplinary laboratory activities.In this context, students are guided through an investigation into the structure of the values of k such that 2k - 1 and 2k + 1 are both prime. Through matrices, modular conditions, and increasingly constraining systems of congruences, an intuitive yet rigorous approach to the Twin Prime Conjecture is constructed, proposing a new didactic perspective on the problem.