Without using Dirichlet's theorem, show that there are infinitely primes congruent to $a \bmod n$ if $a^2\equiv 1(\!\bmod\; n)$. I'd prefer an answer with $Z_{p^2}$ elements if there exists one.
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See this article. I ran into it when doing my homework before asking this question. The upshot there was about limiting the cases where the technique applies. – Jyrki Lahtonen Sep 06 '19 at 16:18
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Mind you, you should try and motivate the question a bit more. See our guide for new askers. – Jyrki Lahtonen Sep 06 '19 at 16:20
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This is a now classical result by Murty:
A Euclidean proof exists for the arithmetic progression $a \bmod n$ iff $a^2 \equiv 1 \bmod n$.
An account can be read in the paper Primes in Certain Arithmetic Progressions by Murty and Thain. See also How I discovered Euclidean proofs by Murty.
See also Euclidean proofs of Dirichlet's theorem by Keith Conrad.
lhf
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