Euler criterion for quadratic residues

Mathematical statement

Require odd primality and a normalized nonzero class first. Compute the Legendre value independently from the complete nonzero square table, and the modular half power by repeated squaring; interpret p−1 as signed−1. Residues give+1 and nonresidues−1. A finite table checks its specific p,a, not the theorem for every odd prime. Modulus2, composite moduli and the zero class are outside this node domain. Do not circularly define Legendre by this criterion or infer a general composite-modulus converse.

Representative formula (exact TeX)

P(a,p)=L(a,p)

Assumptions

• p is a bounded odd prime and a is a normalized safe integer argument.
• Square-table and modular-power computations are separate routes.

Symbol scope

• P — half-exponent residue
• a — integer argument
• p — odd-prime modulus
• L — Legendre symbol

Intuition, definition, and misconceptions

Qualify the domain before applying the formula.

Compare the square table and half exponent without a circular definition.

Keep zero, even-prime and composite boundaries separate.

Do not make the zero class+1, transfer the criterion unchanged to composite moduli or use it to define its own Legendre prerequisite.

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

• Euler's theorem in number theory — Nonzero modulo-p group powers supply background.
• Quadratic residue — Use square classification.
• Legendre symbol — Use its three-valued encoding.

Verification boundary

This slice does not materialize the Jacobi symbol, quadratic reciprocity or general composite square-root decisions.

not_formalized

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

Next obligation

Add quadratic reciprocity and the Jacobi 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.