CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Sums of two squares are closed under products

Statement

The product of two nonnegative integers representable as sums of two squares is again representable as a sum of two squares (Representations and primitive representations as sums of two squares).

Facts & Assumptions

Given: Nonnegative integers $m, n$ , each representable as a sum of two squares.

[F1]

A representation of a nonnegative integer $n$ as a sum of two squares is an ordered pair $(x, y) \in Z^{2}$ such that $n = x^{2} + y^{2}$ (Representations and primitive representations as sums of two squares).

[L1]

For all integers $a, b, c, d$ , $(a^{2} + b^{2}) (c^{2} + d^{2}) = (a c - b d)^{2} + (a d + b c)^{2} = (a c + b d)^{2} + (a d - b c)^{2}$ (The Brahmagupta–Fibonacci two-square identity).

Proof

technique · direct

1.1

Choose integers $a, b, c, d$ with $m = a^{2} + b^{2}$ and $n = c^{2} + d^{2}$ .

givenF1choose

2.1

Then $m n = (a c - b d)^{2} + (a d + b c)^{2}$ by the two-square identity.

step 1.1L1algebra

3.1

The displayed integer pair represents $m n$ by [F1]. This also covers a zero factor, for which the pair $(0, 0)$ represents zero and the same formula gives the zero product.

step 2.1F1∎

Depends on

Used by

Dependency tree · two levels

3 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, Chapter E, §E.II.1 (standard reference, not scraped)
W. Stein, Elementary Number Theory: Primes, Congruences, and Secrets, §5.7 (standard reference, not scraped)