LemmaStatement: 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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The two reciprocity lattice counts partition an open rectangle

Statement

Let $p$ and $q$ be distinct odd primes, and let $S_{p, q}$ and $S_{q, p}$ be the lower-half lattice counts of Gauss's lemma as a lower-half lattice-point count. Then $S_{p, q} + S_{q, p} = (p - 1) (q - 1) / 4$ .

Facts & Assumptions

Given: Distinct odd primes $p, q$ and the integer rectangle $R = {(x, y) : 1 \leq x \leq (p - 1) / 2, 1 \leq y \leq (q - 1) / 2}$ .

[L1]

Put $S_{p, q} : = ∣ {(x, y) \in Z^{2} : 1 \leq x \leq (p - 1) / 2, 0 < p y < q x} ∣$ (Gauss's lemma as a lower-half lattice-point count).

[L2]

If a prime $p$ divides a product $a b$ , then $p ∣ a$ or $p ∣ b$ (Euclid's lemma: if $p$ is prime and $p ∣ a b$ then $p ∣ a$ or $p ∣ b$ ).

Proof

technique · direct

1.1

No point $(x, y) \in R$ lies on the diagonal $p y = q x$ : equality would give $p ∣ q x$ , so [L2] would give $p ∣ q$ or $p ∣ x$ ; distinctness of the primes rules out the first alternative, while $1 \leq x \leq (p - 1) / 2 < p$ rules out the second. Thus every point of $R$ satisfies exactly one of $p y < q x$ and $q x < p y$ .

L2given

2.1

The points of $R$ with $p y < q x$ are exactly those counted by $S_{p, q}$ : the inequality itself forces $y < q / 2$ , hence $y \leq (q - 1) / 2$ ; after interchanging the coordinates and the primes, the points with $q x < p y$ are exactly those counted by $S_{q, p}$ . By step 1.1 these two sets partition $R$ , whose cardinality is $((p - 1) / 2) ((q - 1) / 2) = (p - 1) (q - 1) / 4$ .

step 1.1L1algebra∎

Depends on

Used by

Quadratic reciprocity for distinct odd primes Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 21 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)
A. Gorodnik, Number Theory, Lecture 9, §2 (standard reference, not scraped)