Editing Quadratic reciprocity (section)

===Higher ''q''===
The observations about −3 and 5 continue to hold: −7 is a residue modulo ''p'' if and only if ''p'' is a residue modulo 7, −11 is a residue modulo ''p'' if and only if ''p'' is a residue modulo 11, 13 is a residue (mod ''p'') if and only if ''p'' is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which depend upon congruences modulo 12 and 20 respectively, are simply the ones for −3 and 5 working with the first supplement.

:'''Example.''' For −5 to be a residue (mod ''p''), either both 5 and −1 have to be residues (mod ''p'') or they both have to be non-residues: i.e., ''p'' ≡ ±1 (mod 5) ''and'' ''p'' ≡ 1 (mod 4) or ''p'' ≡ ±2 (mod 5) ''and'' ''p'' ≡ 3 (mod 4). Using the [[Chinese remainder theorem]] these are equivalent to ''p'' ≡ 1, 9 (mod 20) or ''p'' ≡ 3, 7 (mod 20).

The generalization of the rules for −3 and 5 is Gauss's statement of quadratic reciprocity.