 Euclidean proofs of infinitely many primes in each odd residue class modulo four

There are infinitely many primes in each of the classes one and three modulo four. For the first, a hypothetical finite list $p_1,\dots ,p_r$ is contradicted by any prime divisor $q$ of $(2p_1\cdots p_r{)}^2+1$: it is odd, not on the list, and admits a square root of minus one, so $q\equiv 1(\mathrm{mod}4)$. For the second, $4p_1\cdots p_r-1$ is three modulo four; its prime factorization must include a prime that is three modulo four, and none of the listed primes divides it. These special congruence classes therefore admit elementary product-plus-or-minus-one arguments without the general theorem on primes in arithmetic progressions.