Euclidean Proofs of Dirichlet’s Theorem

Euclid’s proof of the infinitude of the primes is a paragon of simplicity: given a finite list of primes, multiply them together and add one. The resulting number, say N , is not divisible by any prime on the list, so any prime factor of N is a new prime. Some special cases of Dirichlet’s theorem admit a simple proof following Euclid’s model, such as the case of 1 mod 4 or 5 mod 6. (We mean by ‘Dirichlet’s theorem’ only the assertion that a congruence class contains infinitely many primes, not the stronger assertion about the density of such primes.) One property which these Euclidean proofs of special cases of Dirichlet’s theorem have in common is that they use a polynomial, varying from one case to the next, whose integer values have restricted prime factors. Specifically, a Euclidean proof of Dirichlet’s theorem for a mod m involves, at the very least, the construction of a nonconstant polynomial h(T ) ∈ Z[T ] for which any prime factor p of any integer h(n) satisfies, with finitely many exceptions, either p ≡ 1 mod m or p ≡ a mod m, and infinitely many primes of the latter type occur. Here are some cases where Euclidean proofs of Dirichlet’s theorem exist.