Assuming that the reader of this preface is an instructor of an actual course, may I first say thank you for introducing your students to number theory! Secondly of course I'm grateful for your at least briefly considering this text.

In that case, gentle reader, you may be asking yourself, “Why on earth yet another undergraduate number theory text?” Surely all of this has been covered in many excellent texts? (See the general preface for a brief topic list, and table of contents for a more detailed one.) And surely there is online content, interactive content, and all the many topics here in other places? Why go to the trouble to write another book, and then to share it? These are excellent questions I have grappled with myself for the past decade.

There are two big reasons for this project. The first is reminiscent of Tertullian's old quote about Athens and Jerusalem; what has arithmetic to do with geometry? (Or calculus, or combinatorics, or anything?) At least in the United States, away from the most highly selective institutions (and in my own experience, there as well), undergraduate mathematics can come across as separate topics connected by some common logical threads, and being at least vaguely about “number” or “magnitude”, but not necessarily part of a unified whole.

When I first taught this course, I was dismayed at how few texts really fully tackled the geometry, algebra, and analysis inherent in number theory. Many do one or two (especially algebra, since number theory might often be a second course in abstract algebra), but few attacked all connections. Still, there are some which do, and even one I found which does a very good job, though at a slightly higher level of sophistication than I found my students ready for.

And I would have happily used this with some extra notes, were it not for the magic and wonder of the internet. How could I not harness this to have my students do approximations to the size of computations that their browsers are constantly doing as they go shopping on the web? Having found Sage, I found it hard to avoid using it whenever I could, and encouraging students to do the same to explore things like Euler's \(\phi\) function.

Interactivity and visualization is becoming common currency in mathematics education. In calculus and lower-level courses this has been true for some time, but even in abstract algebra there are books like Nathan Carter's Visual Group Theory, specialized software projects like PascGalois, and many general applets (including ones from the Wolfram Demonstrations or Maple Möbius projects). Unification.

So my second goal for this book is to bring this into a mainstream number theory text. It is wonderful to see students with an interest in the arts respond to the dynamic visualization in Sage “interacts”, while those with interests in computer science love to ask questions about how to view the source code or some of the details of representing large numbers. And all the students have access to computations from simple ones involving the aliquot parts function to the full Riemann formula for the prime number function.

Why should you not use this book? I make few claims to originality, other than perhaps in the ordering, a few topics I haven't seen addressed adequately very often in truly introductory texts (notably a beginning of the geometry of numbers and long-term averages of arithmetic functions) and some choices of visualization. Likewise, some topics of great importance which are perfect for beginners (especially partitions and continuing fractions) are absent in this first edition. On the other end, I do not and cannot expect a course in abstract algebra or complex (or even real) analysis for my students, and so this book reflects that reality; I have several great recommendations for you if you know all your students can do contour integration or are ready to define a number field. Finally, I don't have a corporation behind me.

On the other hand, this is class-tested material for standard topics (plenty for a semester-long course at most institutions), and not beholden to any interests beyond being a good resource for instructors in “mainstream” undergraduate math programs in the United States. There are plenty of exercises, fun links, and hopefully a quirky and engaging sense of wonder and exploration. The price is also right. Finally, I don't have a corporation behind me.

Should you choose to use this text, I have only a few recommendations for how to use it (similarly to my notes to the student).

• Encourage in-class exploration. Put away books, turn off the computers, and just try stuff out. Create your own worksheet to explore (say) the Möbius function or solutions to linear Diophantine equations. In short, make sure your students see mathematics as a dynamic enterprise – particularly because so many of the theorems involved are highly abstract.
• Less is more. I will often pick one representative proof in a section, project it on the screen, and then really follow it through on an adjacent blackboard with specific numbers (such as \(p=13\), which is just big enough to be interesting but not so big as to be overwhelming).
• Use computer examples judiciously. Sage (or any other system) can just as easily become a Delphic oracle (pun intended) spewing forth cryptic utterances as a useful tool to help create and solve conjectures. You're possibly doing your students a disservice if you don't use it at all (that said, I like this IBL book a lot), but despite having written this text with Sage in mind throughout, I don't regard its use as completely essential. Number theory in this form has been around since Euclid, so the past thirty years of mass-market computation is a drop in the bucket of time.
• Note the Sage notes. Especially if you have more than a couple students who have a little programming experience, this is a perfect course to find projects to challenge them with (such as those in the venerable Rosen number theory text); the Sage notes should gently take you through a short introduction to how to use Python and Sage to do this.

I hope you enjoy using the text as much as I've enjoyed teaching from it. Everyone should have that day where a student's jaw drops from a cool theorem displayed visually, or when the students are working so intently on an in-class project that they don't even notice the class period end. My hope is this text can bring you closer to that goal.