Which cases of Dirichlet's theorem on arithmetic progressions can be proved without analytic tools?
问题内容
I know that Schur and Murty proved that an Euclidean proof (hence a "non-analytic" proof) for the existence of infinite primes $p \equiv \ell \mod q$ with $q$ and $\ell$ coprime can be given if and only if $\ell^2 \equiv 1 \mod q$.
Are there any cases where $\ell^2 \not\equiv 1 \mod q$, but we can find a non-analytic proof of Dirichlet's theorem in this specific case?
The closest thing I found so far, if I recall correctly, is that we can avoid analysis to prove Dirichlet's theorem for sequences $qn+\ell$ for $q$ up to 24. This is contained in Keith Conrad's notes.
It also may be good to specify that by "non-analytic" I mean a proof which uses no analytic concepts such as limits, series and integrals. This also allows for whatever complicated and advanced algebraic argument you may find. Plausibly, I won't be able to understand this kind of proofs, though I'm just curious if there is any way to deviate from classical Euclidean arguments.
(I originally deleted this post because I thought I had found an answer, but actually didn't. Sorry for the inconvenience.)
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