# Quadratic Residues

Canonical: https://manyspace.aporia.systems/topics/quadratic-residues-mth-numth-230
Updated: 2026-10-13

Mathematics · Number Theory · MTH-NUMTH-230

- Difficulty: 3 / 5
- Importance: 4 / 5
- Study time: 4–8 h

Legendre symbol, quadratic reciprocity

## Why it matters

The oldest pure mathematics, now central to cryptography and computer security; also the training ground for abstract algebra.

## Where it is used

RSA and elliptic-curve cryptography, hash functions, error-correcting codes, digital signatures.

## How you know you have learned it

Write 2 independent rigorous proofs + critique/correct one flawed proof.

## What you need first

- [Direct Proof](https://manyspace.aporia.systems/topics/direct-proof-mth-proofs-104) (required)
- [Primitive Roots & Orders](https://manyspace.aporia.systems/topics/primitive-roots-orders-mth-numth-229) (required)
- [Counting Principles](https://manyspace.aporia.systems/topics/counting-principles-mth-disc-112) (helpful)

## Resources

- Elementary Number Theory (Rosen) — Kenneth Rosen