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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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Divisors greater than one of primitively represented integers are primitively represented

Statement

If a positive integer $n$ has a primitive two-square representation and $d > 1$ divides $n$ , then $d$ has a primitive two-square representation.

Facts & Assumptions

Given: A primitively represented positive integer $n$ and a divisor $d > 1$ of $n$ .

[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).

[L2]

For a prime $p$ and a nonzero integer $a$ , $p^{k} ∣ a$ if and only if $k \leq v_{p} (a)$ (For a prime $p$ and a nonzero integer $a$ : $p^{v_{p} (a)} ∣ a$ and $p^{v_{p} (a) + 1} ∤ a$ ; $p^{k} ∣ a$ holds exactly for $k \leq v_{p} (a)$ ; $v_{p} (a) \geq 1$ exactly when $p ∣ a$ ; $v_{p} (1) = v_{p} (- 1) = 0$ ; and $v_{p} (p) = 1$ ).

Proof

technique · direct

1.1

Since $d ∣ n$ , the divisibility criterion in [L2] gives $v_{p} (d) \leq v_{p} (n)$ for every prime $p$ . Thus $v_{2} (d) \leq 1$ , and no three-mod-four prime can divide $d$ , because none divides $n$ by [L1].

givenL1L2

2.1

The two inherited conditions in step 1.1 satisfy [L1], so $d$ has a primitive two-square representation.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

31 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, Corollary E.II.8(b) (standard reference, not scraped)