Quadratic Residues

Is there a number whose square is 2, modulo 7? Yes — 3² = 9 ≡ 2. But modulo 5, no square gives 2: the squares are 1 and 4 only. So 2 is a 'quadratic residue' mod 7 but not mod 5. The question of which numbers are squares mod p sounds simple, but its answer — Gauss's law of quadratic reciprocity — was called 'the golden theorem' and is one of the deepest results in number theory.