It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}$ must lie in certain arithmetic progressions, with a finite number of exceptions. This is because any nonconstant polynomial must have infinitely many distinct prime divisors, which one can prove in a manner imitating Euclid's proof of the infinitude of the primes. For example, taking $p(x) = \Phi_n(x)$, we can prove Dirichlet's theorem for primes congruent to $1 \bmod n$. It is known (see, for example, this paper of K. Conrad) that this is possible precisely for the primes congruent to $a \bmod n$ where $a^2 \equiv 1 \bmod n$.

However, the result about polynomials having infinitely many prime divisors has the following generalization: **any** sequence $a_n$ of integers which is eventually monotonically increasing and which grows slower than $O(2^{\sqrt[k]{n}})$ for every positive integer $k$ has infinitely many distinct prime divisors. In particular, any sequence of polynomial growth (not necessarily a polynomial itself) has this property.

**Question 1:** Given an arithmetic progression $a \bmod n, (a, n) = 1$ such that $a^2 \not \equiv 1 \bmod n$, is it ever still possible to *efficiently* construct a monotonically increasing sequence of positive integers satisfying the above growth condition such that, with finitely many exceptions, the prime divisors of any element of the sequence are congruent to $a \bmod n$? ("Efficiently" rules out answers like "the positive integers divisible by primes congruent to $a \bmod n$," since I do not think it is possible to write down this sequence efficiently. On the other hand, evaluating a polynomial is very efficient.) The idea is that such a sequence immediately gives a proof of Dirichlet's theorem for primes congruent to $a \bmod n$ generalizing the Euclid-style proofs.

**Question 2:** If the above is not possible, are there any known techniques for proving Dirichlet's theorem or at least some of the special cases not covered above without resorting to the usual analytic machinery? For example, Selberg published an "elementary" proof in 1949, but it relies on the "elementary" proof of the prime number theorem, which to me is "finitary analytic machinery." What is the absolute minimum amount of analysis necessary to produce a proof? (Edit: In response to a suggestion in the comments, one way to describe the kind of answer I'm looking for is that it would generalize to a proof of Chebotarev's density theorem that shows very clearly where the distinction between the number field and function field cases is; aside from some "essential" analytic argument there should be no difference between the two.)

This question is inspired at least in part by the following observation: Dirichlet's theorem is equivalent to the seemingly weaker statement that for every progression $a \bmod n, (a, n) = 1$ there exists **at least one** prime congruent to $a \bmod n$. The reason is that if there exists some such prime $a_1$, then letting $n_1$ be the smallest multiple of $n$ greater than $a_1$, there exists a prime congruent to $a_1 + n \bmod n_1$, and so forth.

alwaysfind an efficiently constructed sequence, I'd first like to see you produce just one that is not in the spirit of the Euclid proofs. $\endgroup$7more comments