--- title: "Quadratic Residues and the Law of Reciprocity" description: "Decide which residues are squares modulo a prime, compute the Legendre symbol, and use reciprocity to answer the question quickly." canonical: https://lightmysky.com/learn/mathematics/quadratic-residues-and-the-law-of-reciprocity-mt_0KS9eTCY4p source: https://lightmysky.com/learn/mathematics/quadratic-residues-and-the-law-of-reciprocity-mt_0KS9eTCY4p.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Quadratic Residues and the Law of Reciprocity Decide which residues are squares modulo a prime, compute the Legendre symbol, and use reciprocity to answer the question quickly. Subject: Mathematics · Area: Number Theory · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/quadratic-residues-and-the-law-of-reciprocity-mt_0KS9eTCY4p ## Ready when they can - List the quadratic residues modulo a small prime and count them - Apply Euler's criterion to test whether a residue is a square - Use quadratic reciprocity and the supplementary laws to evaluate a Legendre symbol ## Lesson: Which remainders are squares Quadratic residues ask which remainders occur as squares modulo a prime. For an odd prime p, exactly half of the nonzero remainders are squares, a total of p minus 1 over 2. Modulo 5 the squares are 1 and 4. Modulo 7 they are 1, 2, and 4, since 3 squared is 9 and leaves 2. Pairs a and p minus a always share a square, which is why the list stops at half. Euler criterion reads squareness from one power. Raise your number to half of p minus 1. Remainder 1 means it is a square, remainder minus 1 means it is not. Modulo 7 the power is 3: 2 cubed is 8 and leaves 1, so 2 is a square. One computation replaces a whole search. Reciprocity and its helper laws answer big primes quickly. The helper law for 2 says it is a square exactly when p sits 1 above or 1 below a multiple of 8. Seven sits 1 below 8, so 2 is a square mod 7 with no powering needed. The main flip law swaps a large Legendre question into smaller ones you can finish by hand. **Tip.** List small, power midsize, flip large. Test tiny moduli by squaring bases directly. Use Euler when the power is cheap. Call on reciprocity once the prime grows. Never mistake remainder 1 from random powering for the criterion: only the half power counts. **Recap.** Half the remainders are squares, Euler tests them by powering, and reciprocity shrinks big cases. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Primitive Roots and the Units Modulo n](https://lightmysky.com/learn/mathematics/primitive-roots-and-the-units-modulo-n-mt_KwTDckkMze) ## Opens up - [The Distribution of Primes and What Is Still Open](https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ)