This is a textbook about classical elementary number theory and
elliptic curves. The first part discusses elementary topics such as
primes, factorization, continued fractions, and quadratic forms, in
the context of cryptography, computation, and deep open research
problems. The second part is about elliptic curves, their applications
to algorithmic problems, and their connections with problems in number
theory such as Fermats Last Theorem, the Congruent Number Problem, and
the Conjecture of Birch and Swinnerton-Dyer. The intended audience of
this book is an undergraduate with some familiarity with basic
abstract algebra, e.g. rings, fields, and finite abelian groups.
On November 2008, this book was published by Springer-Verlag and can be purchased from Amazon.com. Springer-Verlag has also very generously agreed to let me make this book completely free online. So please feel free to download it:
Free Legal PDF of this book
Of course, I would greatly appreciate it if you support the book by buying it.
This book is based upon work supported by the National Science Foundation under Grant No. 0653968. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.