CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-17

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.

Jacobi symbol one does not imply a square: $(2 / 15) = 1$

Statement refuted

The converse of A unit square modulo an odd integer has Jacobi symbol one is false: a unit $a$ with $(\frac{a}{n}) = 1$ need not be a square modulo the odd positive integer $n$ .

Facts & Assumptions

Given: The unit numerator $2$ and the odd denominator $15 = 3 \cdot 5$ .

[L1]

For odd $n \geq 1$ with canonical prime factorisation $n = \prod_{i < r} p_{i}^{e_{i}}$ , define $(\frac{a}{n}) : = \prod_{i < r} {(\frac{a}{p_{i}})}^{e_{i}}$ (The Jacobi symbol, with its zero value and empty-product convention).

Counterexample

technique · direct

1.1L1givenalgebra

The nonzero square classes modulo $3$ are ${1}$ and those modulo $5$ are ${1, 4}$ , so $(2 / 3) = - 1$ and $(2 / 5) = - 1$ . Using $15 = 3 \cdot 5$ in [L1] gives $(2 / 15) = (- 1) (- 1) = 1$ .

2.1step 1.1algebra∎

If $x^{2} \equiv 2 (m o d 15)$ , reduction modulo $3$ would give $x^{2} \equiv 2 (m o d 3)$ , but the complete square set modulo $3$ is ${0, 1}$ . Thus $2$ is not a square modulo $15$ , despite its Jacobi symbol being $1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Hackman, Elementary Number Theory, §D.II (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 10, Note 1.3 (standard reference, not scraped)

Jacobi symbol one does not imply a square: (2/15)=1(2/15)=1