HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF Ram Murty

Dirichlet’s 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m = 3 or m = 4), where the proofs parallel Euclid’s argument for the existence of infinitely many primes. It is natural to wonder whether Dirichlet’s theorem in its entirety can be proved by such ‘Euclidean’ arguments. In 1912, Schur showed that one can construct an argument of this type for every progression a mod m satisfying a2 ≡ 1 (mod m), and in 1988 Murty showed that these are the only progressions for which such an argument can be given. Murty’s proof uses some deep results from algebraic number theory (in particular the Chebotarev density theorem). Here we give a heuristic explanation for this result by showing how it follows from Bunyakovsky’s conjecture on prime values of polynomials. We also propose a widening of Murty’s definition of a Euclidean proof. With this defini-tion, it appears difficult to classify the progressions for which such a proof exists. However, assuming Schinzel’s Hypothesis H, we show that again such a proof exists only when a² ≡ 1 (mod m)