by a quadratic form: He did not state the law of quadratic reciprocity, although the cases −1, ±2, and ±3 are easy deductions from these and other of his theorems. He also claimed to have a proof that if the prime number p ends with 7, (in base 10) and the prime number q ends in 3, and p ≡ q ≡ 3 (mod 4), then Euler conjectured, and Lagrange proved, that Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem. Euler Translated into modern notation, Euler stated 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 ≡ b2 (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 ≡ ±b2 (mod 4q). This is equivalent to quadratic reciprocity. He could not prove it, but he did prove the second supplement. Legendre and his symbol Fermat proved that if p is a prime number and a is an integer, Thus if p does not divide a, using the non-obvious fact (see for example Ireland and Rosen below) that the residues modulo p form a field and therefore in particular the multiplicative group is cyclic, hence there can be at most two solutions to a quadratic equation: Legendre lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: He says that since expressions of the form will come up so often he will abbreviate them as: This is now known as the Legendre symbol, and an equivalent definition is used today: for all integers a and all odd primes p Legendre's version of quadratic reciprocity He notes that these can be combined: A number of proofs, especially those based on Gauss's Lemma, explicitly calculate this formula. The supplementary laws using Legendre symbols From these two supplements, we can obtain a third reciprocity law for the quadratic character -2 as follows: For -2 to be a quadratic residue, either -1 or 2 are both quadratic