LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1L2given

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 $q x < p y$ .

2.1step 1.1L1algebra∎

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 $((p - 1) / 2) ((q - 1) / 2) = (p - 1) (q - 1) / 4$ .

Depends on

Used by

Quadratic reciprocity for distinct odd primes 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.V (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 9, §2 (standard reference, not scraped)