PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

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A unit square modulo an odd integer has Jacobi symbol one

Statement

If $a$ is a unit square modulo an odd positive integer $n$ , then $(\frac{a}{n}) = 1$ .

Explicitly, if $\gcd (a, n) = 1$ and $x^{2} \equiv a (m o d n)$ for some integer $x$ , then the Jacobi symbol of $a$ modulo $n$ is $1$ .

Facts & Assumptions

Given: An odd positive integer $n$ and integers $a, x$ such that $\gcd (a, n) = 1$ and $x^{2} \equiv a (m o d n)$ .

[L1]

The assignment $χ_{n} ([u]_{n}) = (\frac{u}{n})$ is a group homomorphism $(Z / n)^{\times} \to {\pm 1}$ (For fixed odd modulus, the Jacobi symbol is a homomorphism on the unit group).

[L2]

A residue class $[u]_{n}$ is a unit if and only if $\gcd (u, n) = 1$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ ).

Proof

technique · direct

1.1

By [L2], $[a]_{n}$ is a unit. If $v$ is its inverse, then $[x]_{n} ([x]_{n} v) = [x]_{n}^{2} v = [a]_{n} v = [1]_{n}$ , so $[x]_{n}$ is also a unit and lies in the domain of [L1].

L1L2givenalgebra

2.1

Applying [L1] to $[a]_{n} = [x]_{n}^{2}$ gives $(\frac{a}{n}) = χ_{n} ([a]_{n}) = χ_{n} ([x]_{n})^{2} = 1$ , since $χ_{n} ([x]_{n}) \in {\pm 1}$ .

step 1.1L1algebra∎

Depends on

Used by

Jacobi symbol minus one obstructs quadratic residuosity Corollary
Jacobi symbol one does not imply a square: (2/15)=1 Counterexample
The kernel of the Jacobi map and the subgroup of unit squares Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Hackman, Elementary Number Theory, §D.II (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 10, §1 (standard reference, not scraped)
V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., §12.2 (standard reference, not scraped)