# reduced integral binary quadratic forms

###### Definition 1.

A *positive* binary form is one where $F\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{\ge}\mathrm{0}\mathit{}\mathrm{\forall}x\mathrm{,}y\mathrm{\in}\mathrm{Z}$.

This article deals only with positive integral binary quadratic forms (i.e. those with negative discriminant^{} and with $a>0$). Some but not all of this theory applies to forms with positive discriminant.

###### Proposition 1.

If $F$ is positive, then $a\mathrm{>}\mathrm{0}$. If $$ and either $a\mathrm{>}\mathrm{0}$ or $c\mathrm{>}\mathrm{0}$, then $F$ is positive.

###### Proof.

If $F$ is positive, then $F(1,0)=a$, so $a>0$.

If $$, then $4aF(x,y)={(2ax+by)}^{2}-\mathrm{\Delta}{y}^{2}\ge 0$. Thus if $a>0$, then $F(x,y)\ge 0$. The proof for the case $c>0$ is identical.
∎

###### Definition 2.

A primitive positive form $a\mathit{}{x}^{\mathrm{2}}\mathrm{+}b\mathit{}x\mathit{}y\mathrm{+}c\mathit{}{y}^{\mathrm{2}}$ is *reduced* if

$$|b|\le a\le c\mathit{\text{, and}}b\ge 0\mathit{\text{if either}}|b|=a\mathit{\text{or}}a=c$$ |

This is equivalent^{} to saying that $\mathrm{-}a\mathrm{\le}b\mathrm{\le}a\mathrm{\le}c$ and that $b$ can be negative only if $$ or $$. Thus $\mathrm{(}\mathrm{3}\mathrm{,}\mathrm{-}\mathrm{2}\mathrm{,}\mathrm{4}\mathrm{)}$ is reduced, but $\mathrm{(}\mathrm{3}\mathrm{,}\mathrm{-}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{)}$ is not.

It turns out that each proper equivalence class^{} of primitive positive forms contains a single reduced form, and thus we can understand how many classes there are of a given discriminant by studying only the reduced forms.

###### Theorem 2.

If $F$ is a primitive positive reduced form with discriminant $\mathrm{\Delta}\mathrm{\ne}\mathrm{-}\mathrm{3}$, $F\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}a\mathit{}{x}^{\mathrm{2}}\mathrm{+}b\mathit{}x\mathit{}y\mathrm{+}c\mathit{}{y}^{\mathrm{2}}$, then the minimum value assumed by $F$ if $x\mathrm{,}y$ are not both zero is $a$. If $$, then this value is assumed only for $\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{\pm}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{)}$; if $a\mathrm{=}c$; it is assumed for $\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{\pm}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{)}$ and $\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{\pm}\mathrm{1}\mathrm{)}$.

###### Proof.

Since $|b|\le a\le c$, it follows that

$$F(x,y)\ge (a-|b|+c)\mathrm{min}({x}^{2},{y}^{2})$$ |

Thus, $F(x,y)\ge a-|b|+c$ whenever $xy\ne 0$, while if $x$ or $y$ is zero, then $F(x,y)\ge a$. So $a$ is clearly the smallest nonzero value of $F$.

If $$, then $F(x,y)\ge a+(c-|b|)>a$ if $xy\ne 0$, and $F(0,y)\ge c>a$ for $y\ne 0$, so $F$ achieves its minimum only at $(x,y)=(\pm 1,0)$.

If $a=c$, then $|b|\ne a$ since otherwise $F$ is not reduced (we cannot have $a=c=1,b=\pm 1$ else we have a form of discriminant $-3$). Thus again $c-|b|>0$ and thus $F(x,y)>a$ if $xy\ne 0$, so in this case the result follows as well. ∎

Note that the reduced form of discriminant $-3$, ${x}^{2}+xy+{y}^{2}$, also achieves its minimum value at $(1,-1),(-1,1)$.

###### Theorem 3.

If $F\mathrm{,}G$ are primitive positive reduced forms with $F\mathrm{\sim}G$, then $F\mathrm{=}G$.

###### Proof.

We take the cases $\mathrm{\Delta}\ne -3$ and $\mathrm{\Delta}=-3$ separately.

First assume $\mathrm{\Delta}\ne -3$ so we can apply the above theorem.

Since $F\sim G$, we can write $G(x,y)=F(\alpha x+\beta y,\gamma x+\delta y)$ with $\alpha \delta -\beta \gamma =1$. Suppose $F=a{x}^{2}+bxy+c{y}^{2},G={a}^{\prime}{x}^{2}+{b}^{\prime}xy+{c}^{\prime}{y}^{2}$. Now, $F$ and $G$ have the same minimum value, so $a={a}^{\prime}$.

If $$, then $F$ achieves its minimum only at $(pm1,0)$, so $a={a}^{\prime}=G(1,0)=F(\alpha ,\gamma )$ and thus $\alpha =\pm 1,\gamma =0$. So $G(x,y)=F(\pm x+ry,\pm y)$ and thus ${b}^{\prime}=b+2ra$. Since $G$ is also reduced, $b={b}^{\prime}$ and thus $c={c}^{\prime}$ and $F=G$.

If instead $a=c$, then instead of concluding that $\alpha =\pm 1$ we can only conclude that $\alpha =\pm 1$ or $\gamma =\pm 1$. If $\alpha =\pm 1$, the argument carries through as above. If $\gamma =\pm 1$, then $\alpha =0,\beta =\mp 1$, so $G(x,y)=F(\mp y,\pm x+ry)$ and thus ${b}^{\prime}=\pm 2cr-b$. Thus ${b}^{\prime}=-b$. But then $c={c}^{\prime}$ since the discriminants are equal, and thus both $b,{b}^{\prime}\ge 0$. So $b={b}^{\prime}=0$ and we are done.

Finally, in the case $\mathrm{\Delta}=-3$, we see that for any such reduced form, $3=4ac-{b}^{2}\ge 4{a}^{2}-{a}^{2}=3{a}^{2}$, so $a=1,b=\pm 1,c=1$. Thus $b=1$ since otherwise the form is not reduced. So the only reduced form of discriminant $-3$ is in fact ${x}^{2}+xy+{y}^{2}$. ∎

###### Theorem 4.

Every primitive positive form is properly equivalent to a unique reduced form.

###### Proof.

We just proved uniqueness, so we must show existence. Note that I used a different method of proof in class that relied on “infinite descent” to get the result in the first paragraph below; the method here is just as good but provides less insight into how to actually reduce a form.

We first show that any such form is properly equivalent to some form satisfying $|b|\le a\le c$. Among all forms properly equivalent to the given one, choose $F(x,y)=a{x}^{2}+bxy+c{y}^{2}$ such that $|b|$ is as small as possible (there may be multiple^{} such forms; choose one of them). If $|b|>a$, then

$$G(x,y)=F(x+my,y)=a{x}^{2}+(2am+b)xy+{c}^{\prime}{y}^{2}$$ |

is properly equivalent to $F$, and we can choose $m$ so that $$, contradicting our choice of minimal $|b|$. So $|b|\le a$; similarly, $|b|\le c$. Finally, if $a>c$, simply interchange $a$ and $c$ (by applying the proper equivalence $(x,y)\mapsto (-y,x)$) to get the required form.

To finish the proof, we show that if $F(x,y)=a{x}^{2}+bxy+c{y}^{2}$, where $|b|\le a\le c$, then $F$ is properly equivalent to a reduced form. The form is already reduced unless $$ and either $a=-b$ or $a=c$. But in these cases, the form $G(x,y)=a{x}^{2}-bxy+c{y}^{2}$ is reduced, so it suffices to show that $F$ and $G$ are properly equivalent. If $a=-b$, then $(x,y)\mapsto (x+y,y)$ takes $a{x}^{2}-axy+c{y}^{2}$ to $a{x}^{2}+axy+c{y}^{2}$, while if $a=c$, then $(x,y)\mapsto (-y,x)$ takes $a{x}^{2}+bxy+a{y}^{2}$ to $a{x}^{2}-bxy+a{y}^{2}$. ∎

Let’s see how to reduce $82{x}^{2}+51xy+8{y}^{2}$ to ${x}^{2}+xy+6{y}^{2}$:

Form | Transformation^{} |
Result |
---|---|---|

$82{x}^{2}+51xy+8{y}^{2}$ | $(x,y)\mapsto (-y,x)$ | $8{x}^{2}-51xy+82{y}^{2}$ |

$8{x}^{2}-51xy+82{y}^{2}$ | $(x,y)\mapsto (x+3y,y)$ | $8{(x+3y)}^{2}-51(x+3y)y+82{y}^{2}=$ |

$8{x}^{2}+48xy+72{y}^{2}-51xy-153{y}^{2}+82{y}^{2}=$ | ||

$8{x}^{2}-3xy+{y}^{2}$ | ||

$8{x}^{2}-3xy+{y}^{2}$ | $(x,y)\mapsto (-y,x)$ | ${x}^{2}+3xy+8{y}^{2}$ |

${x}^{2}+3xy+8{y}^{2}$ | $(x,y)\mapsto (x-y,y)$ | ${(x-y)}^{2}+3(x-y)y+8{y}^{2}={x}^{2}+xy+6{y}^{2}$ |

${x}^{2}+xy+6{y}^{2}$ |

###### Theorem 5.

If $F\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}a\mathit{}{x}^{\mathrm{2}}\mathrm{+}b\mathit{}x\mathit{}y\mathrm{+}c\mathit{}{y}^{\mathrm{2}}$ is a positive reduced form with $$, then $a\mathrm{\le}\sqrt{\frac{\mathrm{|}\mathrm{\Delta}\mathrm{|}}{\mathrm{3}}}$.

###### Proof.

$-\mathrm{\Delta}=4ac-{b}^{2}\ge 4{a}^{2}-{a}^{2}$ since the form is reduced. So $-\mathrm{\Delta}\ge 3{a}^{2}$, and the result follows. ∎

###### Definition 3.

If $$, then *$h\mathrm{}\mathrm{(}\mathrm{\Delta}\mathrm{)}$* is the number of classes of primitive positive forms of discriminant $\mathrm{\Delta}$.

###### Corollary 6.

If $$, then $h\mathit{}\mathrm{(}\mathrm{\Delta}\mathrm{)}$ is finite, and $h\mathit{}\mathrm{(}\mathrm{\Delta}\mathrm{)}$ is equal to the number of primitive positive reduced forms of discriminant $\mathrm{\Delta}$.

###### Proof.

Essentially obvious. Since every positive form is properly equivalent to a (unique) reduced form, $h(\mathrm{\Delta})$ is clearly equal to the number of positive reduced forms of discriminant $\mathrm{\Delta}$. But given a reduced form of discriminant $\mathrm{\Delta}$, there are only finitely many choices for $a>0$, by the proposition^{}. This constrains us to finitely many choices for $b$, since $$. $a$ and $b$ determine $c$ since $\mathrm{\Delta}$ is fixed.
∎

Examples:
$\mathrm{\Delta}=-4$: ${b}^{2}-4ac=-4\Rightarrow b$ even, $|b|\le |a|\le \sqrt{\frac{4}{3}}\Rightarrow b=0$. So $(1,0,1)$, corresponding to ${x}^{2}+{y}^{2}$, is the only reduced form of discriminant $-4$. Note that this provides another proof that primes $\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(4)$ are representable as the sum of two squares, since all such primes have $\left(\frac{-4}{p}\right)=\left(\frac{-1}{p}\right)=1$ and thus are representable by this quadratic form^{}.

$\mathrm{\Delta}=-23$: ${b}^{2}-4ac=-23\Rightarrow b$ odd, $|b|\le a\le \sqrt{\frac{23}{3}}\Rightarrow b=\pm 1$. So $$. This gives us

$(1,1,6)$ | |
---|---|

$(1,-1,6)$ | not reduced since $$; properly equivalent to $(1,1,6)$ via $(x,y)\mapsto (x+y,y)$ |

$(2,1,3)$ | |

$(2,-1,3)$ | reduced since $b|\ne a,a\ne c$ |

There are three equivalence classes of positive reduced forms with discriminant $-23$.

$\mathrm{\Delta}=-55$: $4ac-{b}^{2}=55$, so $b$ is odd, $|b|\le \sqrt{\frac{55}{3}}\Rightarrow |b|=1,\pm 3$. So $ac=14\text{or}16,a\le c$. So the forms are

$(1,1,14)$ | |
---|---|

$(1,-1,14)$ | not reduced since $$, properly equivalent to $(1,1,14)$ via $(x,y)\mapsto (x+y,y)$ |

$(2,1,7)$ | |

$(2,-1,7)$ | reduced since $|b|\ne a,a\ne c$ |

$(4,3,4)$ | |

$(4,-3,4)$ | not reduced since $$, equivalent to $(4,3,4)$ via $(x,y)\mapsto (-y,x)$ |

There are four classes of forms of discriminant $-55$.

$\mathrm{\Delta}=-163$: ${b}^{2}-4ac=-163\Rightarrow b$ odd, $|b|\le |a|\le \sqrt{\frac{163}{3}}\cong \sqrt{55}$. So $b=\pm 1,\pm 3,\pm 5,\pm 7$, and $ac=\frac{{b}^{2}+163}{4}$, so $ac=41,43,45,47$. Since $$, we must have $a=1$; thus $b=\pm 1$ and thus we get only $(1,\pm 1,41)$. But $(1,-1,41)$ is properly equivalent to $(1,1,41)$ via $(x,y)\mapsto (x+y,y)$, so there is only one equivalence class of positive reduced forms with discriminant $-163$.

Title | reduced integral binary quadratic forms |
---|---|

Canonical name | ReducedIntegralBinaryQuadraticForms |

Date of creation | 2013-03-22 19:18:52 |

Last modified on | 2013-03-22 19:18:52 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 4 |

Author | rm50 (10146) |

Entry type | Definition |

Classification | msc 11E12 |

Classification | msc 11E16 |

Related topic | integralbinaryquadraticforms |