theorem quadratic_reciprocity_gauss {p q : ℕ} [Fact (Nat.Prime p)] [Fact (Nat.Prime q)] (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) : legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2))

Gauss's quadratic reciprocity: for distinct odd primes $p, q$, $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\cdot\frac{q-1}{2}}$.