Proof of Quadratic Reciprocity

Number Theory · Axiom Academy

LESSON Proof of Quadratic Reciprocity Using Gauss's Lemma to prove one of number theory's most beautiful theorems The animation shows how we examine multiples of a modulo p. We count how many of the first (p-1)/2 multiples have residues in the upper half (exceeding p/2). This count determines the Legendre symbol. To prove quadratic reciprocity, we'll count lattice points in a specific rectangle. For odd primes p and q, consider the rectangle R with vertices at (0,0), (p/2, 0), (0, q/2), and (p/2, q/2). The animation displays the rectangle divided by the line y = (q/p)x. This line separates the rectangle into two regions, which we'll analyze to apply Gauss's Lemma to both p and q. Let A be the number of lattice points strictly below the line y = (q/p)x in the rectangle, and B be the number strictly above. Since gcd(p,q) = 1, no lattice points lie exactly on the line (except the origin). A equals the count n in Gauss's Lemma applied to (q/p) B equals the count n in Gauss's Lemma applied to (p/q) The animation demonstrates how we count points in each region. The total number of lattice points in the rectangle (excluding axes) is (p-1)/2 × (q-1)/2 = A + B. Since A + B = ((p-1)/2)((q-1)/2) is the total number of interior lattice points:

This is the written version of the interactive lesson above. See the full Number Theory course.