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@\documentclass{article}
\bibliographystyle{amsplain}
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\title{The Triangle Number Problem\\MathCamp 2001}
\author{William A. Stein}
\begin{document}
\maketitle

\section{Lecture 1}
\begin{definition}
A {\em triangle number} is an integer that is the 
area of some right traigle with rational side lengths.
\end{definition}
The area of a right triangle with side lengths $a$, $b$, and $c$
(with $a^2+b^2=c^2$) is $ab/2$.
For example, $a=3$, $b=4$, and $c=5$ are the lengths of the sides
of a right triangle, so $6 = ab/2 = 3\cdot 4/2$ is a triangle number.

\noindent{\bf Big Famous Ancient Problem:}\\
Give a way to decide whether or not a integer~$n$ is a triangle number.

\noindent{\bf Idea!}\\
Let's just list all of the triangles and see what numbers we get.

\subsection{The Greek's method to list the Pythagorean triples}
There is a simple way to list all ``primitive Pythagorean triples'',
that is triples of integers $a,b,c$ with
$a^2+b^2 = c^2$ and $\gcd(a,b,c) = 1$.

Consider the unit circle $x^2 + y^2 = 1$.  
Let $v>u>0$ be integers and draw a line of slope
$u/v$ through $(-1,0)$; this line is defined by the
equation $y = \frac{u}{v}(x+1)$. 

We now find the other point $(x_0, y_0)$ of intersection of
this line and the circle.  
$$x^2 + \frac{u^2}{v^2}(x+1)^2 = 1$$
$$x^2 + \frac{u^2}{v^2}x^2 + \frac{u^2}{v^2}2x + \frac{u^2}{v^2} - 1 = 0$$
$$(u^2 + v^2)x^2 + 2u^2x + u^2 - v^2 = 0$$
$$x^2 + \frac{2u^2}{u^2 + v^2}x + \frac{u^2 - v^2}{u^2 + v^2} = 0$$
Now factor:
 $(x-(-1))(x-x_0) = 0$, so 
$x^2 - (x_0-1)x - x_0=0$, so
$$x_0 = \frac{v^2-u^2}{u^2 + v^2},$$
and 
$$y_0 = \frac{u}{v}\left( \frac{2v^2}{u^2+v^2}\right) = 
        \frac{2uv}{u^2 + v^2}.$$
We have 
$$\left(\frac{v^2-u^2}{u^2+v^2}\right)^2 + 
    \left( \frac{2uv}{u^2+v^2}\right)^2 = 1.$$
Thus 
 $$a = v^2 - u^2, \quad b= 2uv, \quad c=u^2 + v^2$$
form a Pythagorean triple.  Moreover, all primitive
Pythagorean triples arise in this way.
The area of the cooresponding triangle is
$$A = ab/2 = uv(v^2 - u^2).$$

We can now list triangle numbers:


\section{Lecture 2}

\section{Lecture 3}


\bibliography{biblio}
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