Alphabeta Math

Quadratic Residues and the Legendre Symbol — Examples

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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 9 results · all verified · 5 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Quadratic residues, roots, and the Legendre table modulo 11

Example

Fourth and eighth powers modulo 17

Counterexample

Two quadratic nonresidues modulo 15 can have a nonresidue product

Example

The unique cube root of 2 modulo 29 is -3

Example

Euler's criterion computes (3/726377359) by repeated squaring

Example

The signed half-system for 3 modulo 11 gives (3/11)=1

Example

A nonmonic quadratic congruence solved through its discriminant

Counterexample

Squaring is not a homomorphism on a nonabelian group

Counterexample

A soluble square congruence need not define a quadratic residue

Sources