Quadratic Reciprocity Examples

Number Theory · Axiom Academy

EXAMPLE Quadratic Reciprocity Examples Computing Legendre symbols using the Law of Quadratic Reciprocity Excellent work! You've completed this quadratic reciprocity example. Here's what we learned: The Flip Strategy: Quadratic reciprocity lets us "flip" (p/q) to (q/p) when both are odd primes, with a sign correction of (-1)^ (p-1)/2 · (q-1)/2 . Reduction is Key: After flipping, always reduce the numerator modulo the denominator to get smaller, more manageable numbers. Chain of Simplifications: Complex Legendre symbols can be evaluated by repeatedly applying reciprocity and reduction until you reach values you can compute directly (like checking if small numbers are quadratic residues). Congruence Properties: Use the property (a/p) = (b/p) when a ≡ b (mod p) to simplify numerators at each step. Special Cases: Remember that (2/p) has a simple formula, and (-1/p) depends only on whether p ≡ 1 or 3 (mod 4) . This technique is fundamental in number theory for determining quadratic residuosity without direct computation. Practice with different prime pairs to build fluency!

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