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where ''a'', ''b'', ''c'' are the '''coefficients'''. When the coefficients can be arbitrary [[complex number]]s, most results are not specific to the case of two variables, so they are described in [[quadratic form]]. A quadratic form with [[integer]] coefficients is called an '''integral binary quadratic form''', often abbreviated to ''binary quadratic form''.
This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of [[algebraic number theory]]. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to [[quadratic field|quadratic]] and more general [[number field]]s, but advances specific to binary quadratic forms still occur on occasion.
== Equivalence ==
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\alpha \delta - \beta \gamma &= 1\end{align}.</math>
For example, with <math>f= x^2 + 4xy + 2y^2</math> and <math>\alpha = -3</math>, <math>\beta = 2</math>, <math>\gamma = 1</math>, and <math>\delta = -1</math>, we find that ''f'' is equivalent to <math>g = (-3x+2y)^2 + 4(-3x+2y)(x-y)+2(x-y)^2</math>, which simplifies to <math>-x^2+4xy-2y^2</math>.
The above equivalence conditions define an [[equivalence relation]] on the set of integral quadratic forms. It follows that the quadratic forms are [[partition of a set|partition]]ed into equivalence classes, called '''classes''' of quadratic forms. A '''class invariant''' can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class.
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Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant <math>\Delta</math> is '''definite''' if <math>\Delta < 0</math>, '''degenerate''' if <math>\Delta</math> is a perfect square, and '''indefinite''' otherwise. A form is '''primitive''' if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a [[fundamental discriminant]], then the form is primitive.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref> Discriminants satisfy <math>\Delta\equiv 0,1 \pmod 4. </math>
=== Automorphisms ===
If ''f'' is a quadratic form, a matrix
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
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==Representations==
We say that a binary quadratic form <math>q(x,y)</math> '''represents''' an integer <math>n</math> if it is possible to find integers <math>x</math> and <math>y</math> satisfying the equation <math>n = f(x,y).</math> Such an equation is a '''representation''' of ''n'' by ''f''.
=== Examples ===
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</math>
These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form <math>x^2 - 2y^2</math>. This recursive description was discussed in Theon of Smyrna's commentary on [[Euclid's Elements]].
=== The representation problem ===
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The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes. All numbers represented by a definite form <math>f = ax^2 + bxy + cy^2</math> have the same sign: positive if <math>a>0</math> and negative if <math>a<0</math>. For this reason, the former are called '''positive definite''' forms and the latter are '''negative definite'''.
The number of representations of an integer ''n'' by a form ''f'' is finite if ''f'' is definite and infinite if ''f'' is indefinite. We saw instances of this in the examples above: <math>x^2+y^2</math> is positive definite and <math>x^2 - 2y^2</math> is indefinite.
=== Equivalent representations ===
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: <math>\begin{pmatrix} \delta& -\beta \\ -\gamma & \alpha\end{pmatrix} \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}</math>
The above conditions give a (right) action of the group <math>\mathrm{SL}_2(\mathbb{Z})</math> on the set of representations of integers by binary quadratic forms. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
As an example, let <math>f = x^2 - 2y^2</math> and consider a representation <math>1 = f(x_1,y_1)</math>. Such a representation is a solution to the Pell equation described in the examples above. The matrix
: <math> \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix} </math>
has determinant 1 and is an automorphism of ''f''. Acting on the representation <math>1 = f(x_1,y_1)</math> by this matrix yields the equivalent representation <math>1 = f(3x_1 + 4y_1, 2x_1 + 3 y_1)</math>. This is the recursion step in the process described above for generating infinitely many solutions to <math>1 = x^2 - 2y^2</math>. Iterating this matrix action, we find that the infinite set of representations of 1 by ''f'' that were determined above are all equivalent.
There are generally finitely many equivalence classes of representations of an integer ''n'' by forms of given nonzero discriminant <math>\Delta</math>. A complete set of representatives for these classes can be given in terms of ''reduced forms'' defined in the section below. When <math>\Delta < 0</math>, every representation is equivalent to a unique representation by a reduced form, so a complete set of representatives is given by the finitely many representations of ''n'' by reduced forms of discriminant <math>\Delta</math>. When <math>\Delta > 0</math>, Zagier proved that every representation of a positive integer ''n'' by a form of discriminant <math>\Delta</math> is equivalent to a unique representation <math>n = f(x,y)</math> in which ''f'' is reduced in Zagier's sense and <math>x > 0</math>, <math>y \geq 0</math>.<ref>{{harvnb|Zagier|1981||loc=}}</ref> The set of all such representations constitutes a complete set of representatives for equivalence classes of representations.
== Reduction and class numbers ==
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'''Composition''' most commonly refers to a [[binary operation]] on primitive equivalence classes of forms of the same discriminant. One of the deepest discoveries of Gauss which makes this set into a finite [[abelian group]] called the '''form class group''' (or simply class group) of discriminant <math>\Delta</math>. [[Class group]]s have since become one of the central ideas in algebraic number theory. From a modern perspective, the class group of a fundamental discriminant <math>\Delta</math> is [[isomorphic]] to the [[narrow class group]] of the [[quadratic field]] <math>\mathbf{Q}(\sqrt{\Delta})</math> of discriminant <math>\Delta</math>.<ref>{{harvnb|Fröhlich|Taylor|1993|loc=Theorem 58}}</ref> For negative <math>\Delta</math>, the narrow class group is the same as the [[ideal class group]], but for positive <math>\Delta</math> it may be twice as big.
"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.
"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{harvnb|Shanks|1989}} than composition of forms, but arose first historically. We will consider such operations in a separate section below.
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# Compute ''C'' such that <math>\Delta = B^2 - 4AC</math>. It can be shown that ''C'' is an integer.
The form <math>Ax^2 + Bxy + Cy^2</math> is "the" composition of <math>f_1</math> and <math>f_2</math>. We see that its first coefficient is well-defined, but the other two depend on the choice of ''B'' and ''C''. One way to make this a well-defined operation is to make an arbitrary convention for how to choose ''B''
: <math>\begin{pmatrix} 1 & n\\ 0 & 1\end{pmatrix}</math>,
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