Editing Quadratic reciprocity (section)

===Euler===
Translated into modern notation, Euler stated <ref>Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61</ref> that for distinct odd primes ''p'' and ''q'':

# If ''q'' ≡ 1 (mod 4) then ''q'' is a quadratic residue (mod ''p'') if and only if there exists some integer ''b'' such that ''p'' ≡ ''b''<sup>2</sup> (mod ''q'').
# If ''q'' ≡ 3 (mod 4) then ''q'' is a quadratic residue (mod ''p'') if and only if there exists some integer ''b'' which is odd and not divisible by ''q'' such that ''p'' ≡ ±''b''<sup>2</sup> (mod 4''q'').

This is equivalent to quadratic reciprocity.

He could not prove it, but he did prove the second supplement.<ref>Ireland & Rosen, pp. 69&ndash;70. His proof is based on what are now called Gauss sums.</ref>