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

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The representations $221 = 5^{2} + 14^{2} = 11^{2} + 10^{2}$ recover the factors $13$ and $17$

Example

The normalized representations

$221 = 5^{2} + 14^{2} = 11^{2} + 10^{2}$

feed the factorisation construction and recover $221 = 13 \cdot 17$ .

Facts & Assumptions

Given: The displayed representations of $221$ .

[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 an odd integer, two essentially different normalized representations force a factorisation $N = P Q$ with $P, Q > 1$ (Two essentially different two-square representations factor an odd integer).

Verification

technique · direct

1.1

Both equalities are direct, and $(5, 14)$ and $(11, 10)$ are positive odd-even normalized pairs with $0 < 5 < 11$ .

F1algebra

2.1

In the notation of [L1], the values $e = 2$ , $f = 3$ , $g = 4$ , $h = 1$ give $e g - f h = 5$ , $f g + e h = 14$ , $e g + f h = 11$ , and $f g - e h = 10$ .

step 1.1L1algebra

3.1

The resulting factors are $e^{2} + f^{2} = 13$ and $g^{2} + h^{2} = 17$ , and $13 \cdot 17 = 221$ .

step 2.1F1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

P. Hackman, Elementary Number Theory, Chapter E, Theorem E.I.3 and Example E.II.3(a) (standard reference, not scraped)

The representations 221=52+142=112+102221=52+142=112+102 recover the factors 1313 and 1717