TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 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.

The two supplementary laws for the Jacobi symbol

Statement

For every odd positive integer $n$ ,

$(\frac{- 1}{n}) = (- 1)^{(n - 1) / 2}, (\frac{2}{n}) = (- 1)^{(n^{2} - 1) / 8} .$

Both formulas include $n = 1$ , where each Jacobi symbol and each displayed power of $- 1$ equals $1$ .

Facts & Assumptions

Given: An odd positive integer $n$ .

[L1]

The Jacobi symbol is the product of the prime Legendre symbols, taken with the multiplicities in the canonical prime factorisation (The Jacobi symbol, with its zero value and empty-product convention).

[L2]

For odd positive $u, v$ , $(\frac{a}{u v}) = (\frac{a}{u}) (\frac{a}{v})$ (The Jacobi symbol is multiplicative in numerator and denominator).

[L3]

For every odd prime $p$ , $(\frac{- 1}{p}) = (- 1)^{(p - 1) / 2}$ (First supplement: $(- 1 / p) = (- 1)^{(p - 1) / 2}$ ).

[L4]

For every odd prime $p$ , $(\frac{2}{p}) = (- 1)^{(p^{2} - 1) / 8}$ (Second supplement: $(2 / p) = (- 1)^{(p^{2} - 1) / 8}$ ).

Proof

technique · direct

1.1L1L2L3algebra

Expand $(\frac{- 1}{n})$ through [L1], applying [L3] to every prime factor with multiplicity and [L2] to multiply the contributions. For odd $u, v$ , the difference $(u v - 1) / 2 - (u - 1) / 2 - (v - 1) / 2 = (u - 1) (v - 1) / 2$ is even, so iteration through the factor list gives $(\frac{- 1}{n}) = (- 1)^{(n - 1) / 2}$ ; for the empty factor list $n = 1$ , both sides are $1$ .

2.1L1L2L4algebra∎

Similarly, [L1] and [L4] give the product of the signs $(- 1)^{(p^{2} - 1) / 8}$ with multiplicity. For odd $u, v$ , the difference $((u v)^{2} - 1) / 8 - (u^{2} - 1) / 8 - (v^{2} - 1) / 8 = (u^{2} - 1) (v^{2} - 1) / 8$ is even, because each of $u^{2} - 1$ and $v^{2} - 1$ is divisible by $8$ . Iterating this identity gives $(\frac{2}{n}) = (- 1)^{(n^{2} - 1) / 8}$ , again with value $1$ at $n = 1$ .

Depends on

Used by

The Euclidean algorithm computes the Jacobi symbol without factoring the denominator Theorem

Dependency tree · two levels

15 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, §1 (standard reference, not scraped)
V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., §12.2 (standard reference, not scraped)