Number theory is a branch of mathematics which draws its vitality from a rich historical
background. It is also traditionally nourished through interactions with other areas of
research such as algebra algebraic geometry topology complex analysis and harmonic
analysis. More recently it has made a spectacular appearance in the field of theoretical
computer science and in questions of communication cryptography and error-correcting codes.
Providing an elementary introduction to the central topics in number theory this book spans
multiple areas of research. The first part corresponds to an advanced undergraduate course. All
of the statements given in this part are of course accompanied by their proofs with perhaps
the exception of some results appearing at the end of the chapters. A copious list of exercises
of varying difficulty are also included here. The second part is of a higher level and is
relevant for the first year of graduate school. It contains an introduction to elliptic curves
and a chapter entitled Developments and Open Problems which introduces and brings together
various themes oriented toward ongoing mathematical research. Given the multifaceted nature of
number theory the primary aims of this book are to: - provide an overview of the various forms
of mathematics useful for studying numbers - demonstrate the necessity of deep and classical
themes such as Gauss sums - highlight the role that arithmetic plays in modern applied
mathematics - include recent proofs such as the polynomial primality algorithm - approach
subjects of contemporary research such as elliptic curves - illustrate the beauty of arithmetic
The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It
will be of use to undergraduates graduates and phd students and may also appeal to
professional mathematicians as a reference text.
