# Quadratic residue

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p.

mko-quadratic-residue · definition · v0.10.0



## Mathematical statement

Normalize a into0,...,p−1 first. The zero class is handled separately and is not called a nonzero quadratic residue. For odd prime p, every nonzero residue has two distinct opposite roots x and −x. Do not silently include p=2, composite moduli or zero. Euler criterion and the Legendre symbol are not assumed in this node.





## Representative formula (exact TeX)

Q(a,p)





## Assumptions


- Main families enumerate every square of x=1,...,p−1.
- p is a bounded odd prime and a is a safe integer.





## Symbol scope


- Q — quadratic-residue predicate
- a — normalized residue candidate
- p — odd-prime modulus





## Intuition, definition, and misconceptions

The complete nonzero square table decides membership.

Roots pair as x and −x.

Zero, even-prime and composite boundaries remain separate.

Do not count0 among nonzero residues or transfer the two-root conclusion unchanged to composite moduli.





## Adopted derivation guide

[The complete nonzero square table modulo 7](https://ocme.evemisslab.com/en/derivations/derivation-mko-quadratic-residue-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — Use square congruence.
- [Prime natural number](https://ocme.evemisslab.com/en/mko/mko-prime-number/) — The domain requires an odd prime.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The operation is squaring.





## Verification boundary

Euler criterion and the Legendre symbol are reserved for later nodes.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the Legendre symbol.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/quadratic-residue.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-quadratic-residue/index.md)