# Legendre symbol

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

For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively.

mko-legendre-symbol · definition · v0.10.0



## Mathematical statement

Normalize a modulo odd prime p first. If p divides a, (a/p)=0; otherwise the value is1 for a nonzero square class and−1 for another nonzero class. Classification uses a complete nonzero square table or root witness, not a prematurely assumed Euler criterion. Modulus2 and composite moduli are outside this node domain, and a few multiplicativity samples do not prove a general result.





## Representative formula (exact TeX)

L(a,p)=s





## Assumptions


- Positive families enumerate every square of1,...,p−1 for a bounded odd prime.
- The zero class independently gets0; residue/nonresidue classification then concerns normalized nonzero classes only.





## Symbol scope


- L — Legendre symbol
- a — integer argument
- p — odd-prime modulus
- s — three-valued classification0 or plus/minus1





## Intuition, definition, and misconceptions

The normalized zero class is handled before residue classification.

A complete square table supplies both roots and a complete nonresidue absence certificate.

Euler criterion is a later theorem, not a circular definition of this symbol.

Do not give the zero class symbol1, use the same Legendre notation for composite moduli or pre-use Euler criterion to establish its own prerequisites.





## Adopted derivation guide

[The three-valued Legendre symbol from squares](https://ocme.evemisslab.com/en/derivations/derivation-mko-legendre-symbol-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Quadratic residue](https://ocme.evemisslab.com/en/mko/mko-quadratic-residue/) — Use nonzero square classification.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — For fixed p this is a three-valued function.





## Verification boundary

This slice uses no Euler criterion and does not extend the Legendre symbol to the Jacobi symbol.

not_formalized

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





## Next obligation

Add Euler criterion for quadratic residues.





## 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/legendre-symbol.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-legendre-symbol/index.md)