PropositionStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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A positive integer $n$ is primitively represented by some discriminant $Δ$ form exactly when $Δ$ is a square modulo $4 n$

Statement

Let $n$ be a positive integer and let $Δ \in Z$ . Then the following are equivalent:

some integral binary quadratic form of discriminant $Δ$ primitively represents $n$ ;
$Δ$ is a square modulo $4 n$ .

Facts & Assumptions

Given: A positive integer $n$ and an integer $Δ$ .

[F1]

A form primitively represents $n$ when $n = f (r, s)$ for some integers $r, s$ with $\gcd (r, s) = 1$ (Integers represented, and primitively represented, by a binary quadratic form).

[F2]

The discriminant of $(a, b, c)$ is $b^{2} - 4 a c$ (The discriminant of a binary quadratic form).

[F3]

Proper equivalence means substitution by a determinant-one integer matrix, and properly equivalent forms have the same discriminant (Proper equivalence of binary quadratic forms, Proper equivalence preserves discriminant and primitivity of the form).

[L1]

$\gcd (r, s) = 1$ if and only if $r u - s t = 1$ for some integers $t, u$ ( $a$ and $b$ are coprime if and only if $a x + b y = 1$ for some integers $x, y$ ; and in that case the only common divisors of $a$ and $b$ are $1$ and $- 1$ ).

[L2]

$a \equiv b (m o d m)$ means that $m$ divides $a - b$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

Proof

technique · direct

1.1F1L1givenchoose

Suppose a form $f = (a, b, c)$ of discriminant $Δ$ primitively represents $n$ , say $n = f (r, s)$ with $\gcd (r, s) = 1$ . By [L1] choose integers $t, u$ with $r u - s t = 1$ , and put $M = (\begin{array}{cc} r & t \\ s & u \end{array}) \in {S L}_{2} (Z)$ .

1.2F1F2L2givenconstructalgebra

Conversely, suppose $Δ \equiv b^{2} (m o d 4 n)$ for some integer $b$ . Then $4 n$ divides $b^{2} - Δ$ , so $c : = (b^{2} - Δ) / (4 n)$ is an integer. The form $f = (n, b, c)$ has discriminant $Δ$ , and $f (1, 0) = n$ with $\gcd (1, 0) = 1$ , so it primitively represents $n$ .

2.1F2F3step 1.1algebra

The properly equivalent form $g = f ∣ M$ has leading coefficient $g (1, 0) = f (r, s) = n$ , so $g = (n, b^{'}, c^{'})$ for some integers $b^{'}, c^{'}$ . By [F3], $g$ has the same discriminant $Δ$ , hence $Δ = b^{' 2} - 4 n c^{'}$ , which says exactly that $Δ \equiv b^{' 2} (m o d 4 n)$ .

3.1step 2.1step 1.2∎

Step 2.1 proves that primitive representation implies the square congruence, and step 1.2 proves the converse.

Depends on

Used by

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Sources

Andrew Granville, Binary Quadratic Forms, Proposition 4.1 (standard reference, not scraped)

A positive integer nn is primitively represented by some discriminant ΔΔ form exactly when ΔΔ is a square modulo 4n4n