\chapter*{Introduction}   

[Introduction: An essay by the author/editor describing the background
and organization of the book.]

\section*{Factorization}
Do you remember writing whole numbers as products of primes?
For example, 
  $$12=2\times 2\times 3.$$
Can this sort of thing always be done?  Is it really hard or really
easy?  For example, is factoring social security numbers ``trivial''
or hopeless?  In fact, it's trivial; even my wristwatch can do it!!
(Mine might be the only wristwatch in the world that can factor social
security numbers, but that's another story.)  What about bigger
numbers?

These questions are important to your everyday life. If somebody out
there secretly knows how to factor $200$-digit numbers quickly, then
that person could easily read you credit card number and expiration date when
you send it to {\tt amazon.com}.   

\section*{Congruences and Public-key Cryptography}
Two numbers~$a$ and~$b$ are {\em congruent modulo another number~$n$} if 
$a=b+nk$ for some integer~$k$.  That $a$ and $b$ are congruent 
just means you can ``get from~$a$  to~$b$ on the number line'' 
by adding or subtracting lots of copies of~$n$.
For example, $14\con 2\pmod{12}$ since
$14=2+12\cdot 1$.
$$\Z/n\Z = \{\text{ equivalence classes of numbers modulo~$n$ }\}.$$

Your web browser's ``secret code language'' uses arithmetic in
$\Z/pq\Z$ to send messages in broad weblight to
{\tt amazon.com}.  How can this possibly be safe!?  
You will find out exactly what is going on.

\section*{Computers}

Computers make the study of properties of whole numbers vastly more
interesting.  A computer is to a number theorist, like a telescope is
to an astronomer.   It would be a shame to teach an astronomy class
without touching a telescope; likewise, it would be shame to teach
this class without telling you how to look at the integers ``through
the lens of a computer''.   


\section*{Sums of Two Squares}
I will tell you how to decide whether or not your order number is a
sum of two squares.  For example, an odd prime number is a sum of two
squares if and only if when divided by~$4$ it leaves a remainder
of~$1$.  For example, $7$ is not a sum of two squares, but $29$ is.

\section*{Elliptic Curves}
My experience is that elliptic curves are extraordinarily fun to
study.  Every such curve is like a whole galaxy in itself, just like
the rational numbers are.
An
elliptic curve over~$\Q$ is a curve that can be put in the form
  $$y^2 = x^3 + ax+b,$$
where the cubic has distinct roots and $a, b\in \Q$.
The amazing thing is that the set of pairs
$$E(\Q) = \{ (x,y)\in \Q\cross \Q : y^2 = x^3 + ax +b \} \union \{\infty\}$$
has a natural structure of ``group''.  In particular,
this means that given two points on $E$, there is a way
to ``add'' the two solutions together to get
another solution.

Many exciting problems in number theory can be translated
into questions about elliptic curves.  
For example, Fermat's Last Theorem, which asserts that 
$x^n + y^n = z^n$ has no positive integer solutions when $n>2$
was proved using elliptic curves.  Giving a method to decide
which numbers are the area of a right triangle with rational
side lengths has {\em almost}, but not quite, been solved using
elliptic curves.

A central question about elliptic curves is {\em The Birch and
Swinnerton-Dyer Conjecture} which gives a simple conjectural criterion
to decide whether or not $E(\Q)$ is infinite (and more).  Proving the
BSD conjecture is one of the Clay Math Institute's million dollar
prize problems.  We will discuss this conjecture in Chapter~\ref{}.




