TheoremStatement: 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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A prime congruent to $1$ modulo $4$ has one two-square representation up to signs and order

Statement

Let $p \equiv 1 (m o d 4)$ be prime. There are unique positive integers $x, y$ with $x$ odd, $y$ even, and

$p = x^{2} + y^{2} .$

Every ordered signed representation of $p$ is obtained from this pair by changing signs and interchanging coordinates (Representations and primitive representations as sums of two squares).

Facts & Assumptions

Given: A prime $p \equiv 1 (m o d 4)$ .

[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]

A prime $p$ is a sum of two integer squares if and only if $p = 2$ or $p \equiv 1 (m o d 4)$ (Fermat's two-square theorem for primes).

[L2]

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).

Proof

technique · direct

1.1

By [L1], choose integers $a, b$ with $p = a^{2} + b^{2}$ .

givenL1F1choose

1.2

Neither coordinate is zero, since a prime cannot be a square of an integer greater than one. Since $p$ is odd, exactly one coordinate is odd: two odd squares sum to $2$ modulo $4$ , and two even squares give an even sum. Changing signs and interchanging coordinates therefore turns every representation into a positive normalized one.

givenF1algebra

2.1

If two normalized representations differed, their positive odd coordinates would differ; order them as $0 < x < u$ and apply [L2]. This would write the prime $p$ as a product of two integers greater than one, a contradiction. Hence the normalized representation is unique.

step 1.2L2given

3.1

Step 1.1 supplies the normalized pair, step 2.1 makes it unique, and undoing the sign changes and interchange in step 1.2 gives all ordered signed representations and no others.

step 1.1step 1.2step 2.1F1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Theorem E.I.3 (standard reference, not scraped)

A prime congruent to 11 modulo 44 has one two-square representation up to signs and order