Binary quadratic form

A binary quadratic form is $a{x}^2+bxy+c{y}^2$; integral changes of variables organize forms into equivalence classes.

For odd prime $p$ and an integral-form discriminant $d$, representation of $p$ by some form of discriminant $d$ is equivalent to solvability of $b^2\equiv d(\mathrm{mod}p)$. A primitive representing vector can be completed to a unimodular change giving leading coefficient $p$. Conversely choose $b$ with the required parity and use $[p,b,(b^2-d)/(4p)]$.

Integral forms $f$ and $g$ are properly equivalent whenfor some $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$. Proper equivalence preserves both represented integers and orientation.

An integral binary quadratic form $[a,b,c]$ is primitive when $\gcd (a,b,c)=1$. This property is invariant under proper equivalence because a unimodular change and its integral inverse express either coefficient triple as integer combinations of the other.

The discriminant of $[a,b,c]$ is $D=b^2-4ac$. It is invariant under proper equivalence, and a real form with $a>0$ is positive definite exactly when $D<0$.

A positive definite integral form $[a,b,c]$ is reduced when $|b|\le a\le c$, with $b\ge 0$ if $|b|=a$ or $a=c$. Every proper equivalence class has a unique reduced representative under this boundary convention.

For a positive definite integral form $[a,b,c]$, a unimodular shear can replace $b$ by its representative $b^{\prime }$ with $|b^{\prime }|\le a$. If the resulting $c^{\prime }=(b^{\prime 2}-D)/(4a)$ is smaller than $a$, swapping the variables replaces the leading coefficient by the smaller positive integer $c^{\prime }$. Repeating terminates and produces $|b|\le a\le c$; determinant-one sign changes enforce the boundary convention. Thus every positive definite integral binary quadratic form is properly equivalent to a reduced form.

For a reduced form of negative discriminant $D$, the inequalities imply $3a^2\le |D|$. One can therefore enumerate all classes by checking the finite setand retaining integral $c=(b^2-D)/(4a)$ with $a\le c$ and the boundary sign convention.

The reduced positive definite triples are $(1,0,11)$, $(2,2,6)$ and $(3,\pm 2,4)$. The imprimitive middle form represents only even numbers; the two last forms have identical value sets after reversing one variable.

The two reduced positive definite integral forms of discriminant $-35$ areFor a prime $p>7$, the first represents $p$ exactly whenand the second represents $p$ exactly when both Legendre symbols are $-1$.

The class number problem asks for number fields or discriminants having a prescribed class number.

For a negative discriminant $D\equiv 0$ or $1(\mathrm{mod}4)$, the class number $h(D)$ is the number of proper equivalence classes of primitive positive definite integral binary quadratic forms of discriminant $D$. Equivalently, it is the number of reduced primitive forms of discriminant $D$.

For an odd prime number $q$, the primitive positive definite formshave discriminant $-8q$ and lie in distinct proper equivalence classes. The first represents one, whereas the second has minimum positive value two.

If $q\equiv 3(\mathrm{mod}8)$ is prime and $h(-8q)=2$, then for every prime $p>q$,for some integers $x,y$ exactly when $p\equiv \pm 1(\mathrm{mod}8)$ and $p$ is a quadratic residue modulo $q$.

Let $d$ be odd and have $k$ distinct prime divisors. Assigning each complete prime-power factor of $d$ to one of two factors gives $2^k$ ordered coprime factorizations $d=ac$. Interchanging $a$ and $c$ identifies pairs, and each unordered factorization gives a different reduced primitive form $[\min (a,c),0,\max (a,c)]$ of discriminant $-4d$. Hence

The identitycomes from an integer change-of-variable matrix of determinant $-1$. It therefore preserves the set of integers represented and reduces the form to $u^2+v^2$.