This is a textbook about prime numbers, congruences, basic public-key cryptography, quadratic reciprocity, continued fractions, elliptic curves, and number theory algorithms. We will mention groups, rings, and fields in passing at various places in this book, but much of the book should be comprehensible without a course in algebra. This book grew out of an undergraduate course that the author taught at Harvard University in 2001 and 2002, at UC San Diego in 2005, and at University of Washington in 2007.
Notation and Conventions. We let
denote the natural numbers, and use the standard
notation
,
,
, and
for the rings of integer,
rational, real, and complex numbers, respectively. In
this book we will use the words proposition, theorem, lemma, and
corollary as follows. Usually a proposition is a less important or
less fundamental assertion, a theorem a deeper culmination of ideas, a
lemma something that we will use later in this book to prove a
proposition or theorem, and a corollary an easy consequence of a
proposition, theorem, or lemma.
Acknowledgements. Brian Conrad and Ken
Ribet made a large number of clarifying comments and suggestions
throughout the book. Baurzhan Bektemirov, Lawrence Cabusora, and
Keith Conrad read drafts of this book and made many comments.
Frank Calegari used the course when teaching Math 124 at Harvard,
and he and his students provided much feedback.
Noam Elkies made comments and suggested
Exercise 4.6. Seth Kleinerman wrote a version of
Section 5.3 as a class project.
Samit Dasgupta, George Stephanides,
Kevin Stern, Ting-You Wang, and Heidi
Williams all suggested corrections. I also benefited from
conversations with Henry Cohn and David Savitt.
I used SAGE, emacs, and LATEX in the preparation of this book.
Numerous mistakes and typos corrected by: Arthur Patterson, Eve Thompson
William 2007-06-01