TheoremStatement: 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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Quadratic reciprocity for distinct odd primes

Statement

For distinct odd primes $p, q$ , $(\frac{p}{q}) (\frac{q}{p}) = (- 1)^{(p - 1) (q - 1) / 4}$ .

Equivalently, the two Legendre symbols agree unless $p \equiv q \equiv 3 (m o d 4)$ , in which case they have opposite signs.

Facts & Assumptions

Given: Distinct odd primes $p$ and $q$ .

[L1]

For distinct odd primes $p, q$ , the lower-half count satisfies $(\frac{q}{p}) = (- 1)^{S_{p, q}}$ (Gauss's lemma as a lower-half lattice-point count).

[L2]

For the two orientations of the rectangle, $S_{p, q} + S_{q, p} = (p - 1) (q - 1) / 4$ (The two reciprocity lattice counts partition an open rectangle).

Proof

technique · direct

1.1

Applying [L1] in both orientations, multiplying, and then using [L2] gives $(\frac{p}{q}) (\frac{q}{p}) = (- 1)^{S_{q, p} + S_{p, q}} = (- 1)^{(p - 1) (q - 1) / 4}$ .

L1L2algebra

2.1

The exponent is $((p - 1) / 2) ((q - 1) / 2)$ , which is odd exactly when both factors are odd, equivalently when $p \equiv q \equiv 3 (m o d 4)$ . Since the Legendre symbols are signs for distinct primes, their product is then $- 1$ , and in every other case it is $1$ , proving the equivalent formulation.

step 1.1algebra∎

Depends on

Used by

A complete reciprocity table for 3,5,7,11 Example
Odd primes represented by a divisor of x²+3 Example
Quadratic reciprocity for coprime odd Jacobi denominators Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 15 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.V (standard reference, not scraped)
W. Stein, Elementary Number Theory, §§4.1 and 4.3 (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 9, §2 (standard reference, not scraped)