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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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$68 = 8^{2} + 2^{2}$ is representable but has no primitive two-square representation

Statement refuted

Every integer representable as a sum of two squares has a primitive two-square representation. The integer $68$ is a counterexample.

Facts & Assumptions

Given: The integer $68$ .

[F1]

A two-square representation is primitive when its coordinate gcd is $1$ (Representations and primitive representations as sums of two squares).

[L1]

A positive integer $n$ has a primitive two-square representation if and only if $v_{2} (n) \leq 1$ and no prime $q \equiv 3 (m o d 4)$ divides $n$ (Characterisation of primitive sums of two squares).

Counterexample

technique · direct

1.1

Directly, $68 = 8^{2} + 2^{2}$ , but $\gcd (8, 2) = 2$ , so the displayed representation is not primitive.

F1algebra

1.2

In any equality $4 k = x^{2} + y^{2}$ , square residues modulo $4$ force both $x$ and $y$ even. Thus no representation of a multiple of four can be primitive.

F1algebra

2.1

Since $v_{2} (68) = 2$ , [L1] gives the same conclusion: $68$ has no primitive representation despite step 1.1.

step 1.1step 1.2L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Example E.II.7(d) (standard reference, not scraped)

68=82+2268=82+22 is representable but has no primitive two-square representation