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An integer is the discriminant of an integral binary quadratic form exactly when it is congruent to $0$ or $1$ modulo $4$

Statement

An integer $Δ$ is the discriminant of an integral binary quadratic form if and only if

$Δ \equiv 0 (m o d 4) or Δ \equiv 1 (m o d 4) .$

Facts & Assumptions

Given: An integer $Δ$ .

[F1]

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

[L1]

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

[L2]

Congruent integers may be added, subtracted, and multiplied (Congruent integers may be added, subtracted and multiplied: representative changes preserve both arithmetic operations).

Proof

technique · direct

1.1

Suppose $Δ$ is the discriminant of some integral form, say $Δ = b^{2} - 4 a c$ . Then $Δ \equiv b^{2} (m o d 4)$ .

F1L1L2

1.2

If $Δ \equiv 0 (m o d 4)$ , then $(1, 0, - Δ / 4)$ is an integral binary quadratic form and its discriminant is $0^{2} - 4 \cdot 1 \cdot (- Δ / 4) = Δ$ .

F1L1givenconstructalgebra

1.3

If $Δ \equiv 1 (m o d 4)$ , then $(1, 1, (1 - Δ) / 4)$ is an integral binary quadratic form and its discriminant is $1^{2} - 4 \cdot 1 \cdot ((1 - Δ) / 4) = Δ$ .

F1L1givenconstructalgebra

2.1

If $b = 2 m$ is even, then $b^{2} = 4 m^{2} \equiv 0 (m o d 4)$ ; if $b = 2 m + 1$ is odd, then $b^{2} = 4 m (m + 1) + 1 \equiv 1 (m o d 4)$ . Hence every discriminant is congruent to $0$ or $1$ modulo $4$ .

step 1.1algebra

3.1

Step 2.1 proves the forward implication, while steps 1.2 and 1.3 prove the converse in the two possible congruence classes.

step 2.1step 1.2step 1.3∎

Depends on

Used by

The principal binary quadratic form of a discriminant Definition

Dependency tree · two levels

9 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

William Stein, Elementary Number Theory and Elliptic Curves, Proposition 9.2.9 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)

An integer is the discriminant of an integral binary quadratic form exactly when it is congruent to 00 or 11 modulo 44