CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Squarefree sums of two squares

Statement

A squarefree positive integer is a sum of two squares if and only if none of its odd prime factors is congruent to $3$ modulo $4$ ; every such representation is primitive.

Facts & Assumptions

Given: A squarefree positive integer $n$ .

[F1]

A positive integer $n$ is squarefree if no square of a prime divides $n$ ; equivalently, every exponent in its canonical prime factorisation is $0$ or $1$ (Squarefree positive integers).

[L1]

A positive integer $n$ is a sum of two squares if and only if every prime $q \equiv 3 (m o d 4)$ occurs to an even exponent in its canonical prime factorisation (Characterisation of positive integers that are sums of two squares).

[L2]

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).

[F2]

An integer $d$ is a common divisor of $a$ and $b$ when $d ∣ a$ and $d ∣ b$ (Common divisor, and the greatest common divisor $\gcd (a, b)$ , with the convention $\gcd (0, 0) : = 0$ ).

[L3]

Every integer greater than $1$ has a prime divisor (Every integer $n > 1$ has a prime divisor; indeed the least divisor of $n$ that exceeds $1$ is prime).

[L4]

If a prime $p$ divides $a b$ , then $p ∣ a$ or $p ∣ b$ (Euclid's lemma: if $p$ is prime and $p ∣ a b$ then $p ∣ a$ or $p ∣ b$ ).

Proof

technique · direct

1.1F1L1algebra

By [F1], every prime exponent of $n$ is zero or one. Consequently the even-exponent condition in [L1] for a prime congruent to three modulo four is equivalent to that prime not dividing $n$ .

1.2F1F2L3L4assume-contraalgebradischarge-contradiction

To see that every representation is primitive, suppose $n = x^{2} + y^{2}$ and $\gcd (x, y) > 1$ . By [L3] choose a prime $ℓ$ dividing the gcd. Then $ℓ ∣ x$ and $ℓ ∣ y$ by [F2], so $ℓ^{2} ∣ n$ , contradicting [F1].

2.1step 1.1L2F1

Squarefreeness also gives $v_{2} (n) \leq 1$ , so the same exclusion of three-mod-four primes satisfies [L2] and yields a primitive representation whenever $n$ is represented.

3.1step 1.1step 2.1step 1.2∎

Step 1.1 proves the representation criterion, step 2.1 supplies primitivity under that criterion, and step 1.2 shows that every representation has it.

Depends on

Used by

Squarefree sums of two squares up to 30 Example

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Sources

P. Hackman, Elementary Number Theory, Chapter E, Theorems E.II.2 and E.II.4 (standard reference, not scraped)