I recently encountered a result which seems to be analogous to the following result of Dirichlet, which I wrote in a few common forms to be more suggestive.

**Theorem (Dirichlet on Arithmetic Progressions).** Let a and b be relatively prime positive integers, then there are infinitely many primes p $\equiv$ a mod b (i.e., the progression $a + b \mathbb{N}$ contains infinitely many primes).

**Corollary 1.** There are infinitely many primes such that a is not a square and quadratic residue (mod p) and infinitely many primes such that a is a quadratic non-residue (mod p). In nicer terms, given n, there are infinitely many primes such that the Legendre symbol $(\frac{-n}{p})$ takes on each value for infinitely many p (for p ≠ 2).

**Corollary 2. **Something like: Given a prime $p$, there’s a fifty-fifty chance that the image of the prime splits (through an order two extension) [remains inert/ramify when passed to the extension]. This is saying half are quadratic non-residues and half are quadratic residues, respectively, that is, half are squares mod p, and the other half aren’t. For example, $2 \mod 5$ is not a square, but $1 \mod 5$ and $3 \mod 5$ are squares.

A result that tastes like Corollary 2 is the existence of infinitely many super singular primes for every elliptic curve over $\mathbb{Q}$: Continue reading A question on primes