Legendre symbol

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 original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Quadratic residue — Use nonzero square classification.
• 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.