so i fouind this website from a guy who is way way way too fanatic about david hilbert and gauss. and he phrased gauss's "golden theorem" of quadratic residues a bit differently than others: https://archive.lib.msu.edu/crcmath/math/math/q/q044.htm
If
pandqare distinct odd primes, then the congruence:
x^2 = q (mod p)x^2 = p (mod q)are both solvable or both unsolvable unless both
pandqleave the remainder 3 when divided by 4 (in which case one of the congruences is solvable and the other is not).
