Textbook in PDF format The study of number theory is a great journey into the recesses of one’s mind. I have attempted to capture some of the essence of this vast historical adventure while at the same time pointing out routes yet uncharted. Like other adventures, it is a personal one. The choice of which theorems to contemplate, exercises to solve, sections to study, and unsolved problems to investigate is all up to you. The success of my book will depend on how coherent a story I have told and whether I have created a smooth and pleasant journey. The book can serve as a textbook or source book in a wide variety of settings from an undergraduate course on number theory, as a primer for computer scientists needing some number theoretic background, to students working on senior theses or master’s projects in mathematics and beyond. It contains a thoroughly up-to-date exposition of elementary number theory together with a host of advanced topics depending on the instructor’s and student’s needs. There are abundant examples throughout to illuminate how the principles actually apply in practice. Many examples are stated as problems with solutions. I hope this encourages students to take a more active role in solving the problems themselves. And I have included reference to many unsolved problems that might entice the interested student to investigate further. There are many historical and biographical remarks interwoven throughout the text to highlight that much of mathematics is a very active human creation to which we can all play a part. Theorems and proofs include accurate attributions to the best of my ability, and I have given descriptive names to many results to aid in their retention. Give credit where it’s due! The proofs are as direct as possible without requiring much beyond a solid background in calculus plus linear algebra. Only minimal familiarity with advanced analytic methods or abstract algebra is required. Over 800 exercises in all of varying depth and difficulty are included. Generally, the easier and more direct ones come first before those that require deeper analysis. Additionally, computational problems tend to precede more theoretical ones. I hope a good balance has been maintained. Rather than starring what I consider to be the difficult problems, let the students discover their own strengths and weaknesses. I expended as much care and effort in creating interesting problems as in writing the text itself