Some remarks on the theorem about the infinity of prime numbers

The famous Euclid’s theorem on the infinity of prime numbers represents a typical case of difficulties for students. In this work we present some reflections and proposals to contrast such difficulties, focused on: a) the problem of proofs by contradiction – in this case viewed as inessential – also in relation with the dychotomy potential/actual infinite; b) a comparison between the current proof and the original Euclid’s one, especially for its potential influence on the building of algebraic language; c) the opportunity of privileging students’ exploratory activities as necessary steps toward the construction of the proof, and the chances that a wise use of technologies offer to this exploration