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I need help interpreting this passage from page 35 of the book Diophantine Analysis by Robert D Carmichael.

"Thus the problem of solving Eq. (1) is reduced to that of solving Eq. (4) for $u$ and $v$ and choosing one of those values for which x and y have the desired characteristic. But the problem of solving Eq. (4) is identical with that of the representation of a given integer by means of a binary quadratic form. The plan of this book does not permit the detailed development of this latter subject."

Equation 1 is $ax^2+2bxy+cy^2+2dx+2ey+f=0$ and equation 4 is $au^2+2buv+cv^2=m$. $u,v,$ and $m$ are defined elsewhere if that matters, but what I need to know is what that bold part is referring to; and that last sentence gives me little reason to believe I can find context in the book to make sense of it.

This is the main part of the book I'm interested in, but I do intend to go through most of it. It wasn't easy to find a book on the subject and I just checked it out from the library today.

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    If we define $f(u,v) = au^2 + 2b uv + cv^2$, then $f$ is a [binary quadratic form](http://en.wikipedia.org/wiki/Binary_quadratic_form) (a homogeneous polynomial of degree 2 in two variables). So solving $f(u,v)=m$ is equivalent to the question "what integers are in the image of $f$?" (that is, what integers can be 'represented' by the form $f$?). This is a classic subject that has roots in work of Fermat and of [Brahmagupta](http://en.wikipedia.org/wiki/Brahmagupta%27s_identity).2012-03-05

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A binary quadratic form is a homogeneous polynomial of degree 2 in two variables, $q(x,y) = ax^2 + bxy + cy^2.$

When $a,b,c\in\mathbb{Z}$, we can view $q$ as a function $q\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$, a common question asks for a description of the elements of the image. For instance, there is the classic problem of determining which integers can be written as the sum of two squares; this can be viewed as asking for the image of the binary quadratic form $q(x,y) = x^2+y^2$ (that is, $a=c=1$, $b=0$). The classic terminology is that an integer $m$ is "represented" by the binary quadratic form $q$ if and only if there exist integers (sometimes positive integers) $r$ and $s$ such that $q(r,s)=m$.

This subject has deep roots; they go back to Fermat (who described all primes that can be represented by $q(x,y)=x^2+y^2$, $q(x,y) = x^2+2y^2$, and $q(x,y)=x^2+3y^2$), Euler (who proved Fermat's assertions and then attempted to extend them to other quadratic forms $q(x,y) = x^2+dy^2$ with $d$ squarefree), Lagrange, and Gauss (who dedicates the entire Part V of his Disquisitiones Arithmeticae to the study of these functions). They also reach into the East: Brahmagupta's identity, for example, shows that if $m$ and $n$ can both be represented by the quadratic form $q(x,y) = x^2+cy^2$, then so can $mn$ (that is, the set of representable integers is closed under products).

What the book is saying is that, via a change of variable, solving the general two-variable quadratic diophantine equation $ax^2 + bxy + cy^2 + dx + ey + f = 0\tag{1}$ is equivalent to the question of determining when a certain integer $m$ is representable by a particular binary quadratic form (where $m$ and the quadratic form depends on $a$, $b$, $c$, $d$, $e$, and $f$); in the sense that solving the diophantine equation $(1)$ yields a representation of $m$; and a representation of $m$ gives a solution to $(1)$.

The subject of quadratic forms is rich and interesting; whole books have been written on the subject by great minds; the Disquisitiones spends more time on them than on everything else put together. Whether an integer can be represented by a binary quadratic form is intimately connected with the arithmetic of quadratic fields. There is a beautiful [Theorem of Conway and Schneeberger] that says that and integral quadratic form with integer matrix that represents the positive integers between 1 and 15 will, in fact, represent all integers. And much more.

But because it is rich and interesting, it is too complicated to get into in any detail in a book that is not devoted to them. Hence, your book is saying that this is a little too hard to get into in detail.

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The point is that, by a linear change of variables, one can normalize the non-homogeneous equation to homogenous form. In fact one can go normalize to Pell form - see below. Such normalization is important since it brings to the fore the innate linearity - making it easier to exploit the relationship between binary quadratic forms and ideals or modules - see below.

Lagrange showed how to reduce a general binary quadratic Diophatine equation to Pell form.

$\rm a\ x^2 + b\ xy + c\ y^2 + d\ x + e\ y + f\ =\ 0 $

reduces to a Pell equation as follows: put $\rm\ D = b^2-4ac,\ E = bd-2ae,\ F = d^2-4af\:.\ $ Then

$\rm D\ Y^2\ =\ (D\ y + E)^2 + D\ F - E^2,\quad\quad Y\ =\ 2ax + by + d $

Therefore if we put $\rm\quad\ \ X\: =\: D\ y + E,\quad\ \ N\: =\: E^2 - D\ F\quad\ \ $ we obtain the Pell equation

$\rm X^2 - D\ Y^2\ =\ N $

Below is a proof of the standard equivalences between forms, ideals and numbers, excerpted from section 5.2, p. 225 of Henri Cohen's book "A course in computational algebraic number theory".

alt text alt text

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    I don't see either of those titles available at any libraries I have access to. I did find something on ideals in my brother's old textbook, A First Course in Abstract Algebra by John B. Fraleigh. I may not have had as in depth a course as he did, but I did help with a lot of the homework, so hopefully it will all come back quickly. But what exactly does SL$_2$ signify?2012-03-06