= Euclidean proofs of infinitely many primes in each odd residue class modulo four
{c}
There are infinitely many primes in each of the classes one and three modulo four. For the first, a hypothetical finite list $p_1,\ldots,p_r$ 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 $q\equiv1\pmod4$. 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.
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