Without using Dirichlet's theorem, show that there are infinitely primes congruent to $a$ mod n if $a^2\equiv 1\,(\!\bmod\; n)$

Question

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.

Answer 1

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.