Preface To the Instructor
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, โWhy on earth yet another undergraduate number theory text?โ Surely all of these topics have been covered in many excellent texts? (See the preface To Everyone for a brief topic list, and the 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 โnumberโ or โmagnitudeโ, 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 I even found Elementary Number Theory by Jones and Jones [E.2.1] which does a very good job of this, though at a slightly higher level of sophistication than I found my students ready for. Those familiar with it will find that my presentation of certain topics (e.g. arithmetic functions, the zeta function) and some topic order is influenced by it; for certain proofs (especially in Dirichlet series) the proofs there and in [E.4.6] are the only ones I could find! I try to point out all such cases, and I have substantively modified even those in ways more appropriate for typical US undergraduates, as well as with somewhat different emphases. Given my first goal, I would have happily used that text with some extra details for my students, 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'sEncourage in-class exploration. Put away books, turn off the computers, and just try stuff out. Create your own worksheet to explore (say) the Mรถbius function or solutions to linear Diophantine equations. In short, make sure your students see mathematics as a dynamic enterprise โ 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
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, 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. If you want a true inquiry-based approach, I like the text Number Theory through Inquiry [E.2.5] a lot.
Note the Sage notes (full list at the List of Sage notes). Especially if you have more than just a few 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 [E.2.4]. The Sage notes gently remind or give short introductions to some aspects of how to use Sage and Python (the language Sage is based on). They are not formally structured or arranged, or comprehensive; if you are looking for this, you should supplement your course with a real basic programming text in Python, such as [E.3.7] or [E.3.8]. (The already-initiated should note that as of January 2020 this book has been updated to Sage 9 and Python 3, so some commands, especially those involving
print()
, may not work with certain earlier versions of Sage.)Use the exercises, and ones outside the book if you want. There are exercises for each chapter, of varying difficulty levels (in the grand tradition of upper-level math texts, I do not provide solutions). In general, assigning daily, collecting weekly seems to be a decent model โ though be sure to give students ample warning as to which ones will be collected! The last few chapters' material is more advanced, and there are correspondingly fewer possible exercises. I find this to be a good time for a small project in the history of number theory; especially if you have students from several different cultural heritages, having them discover where it comes up in theirs (it nearly always does) has been a perennial favorite.
If you are teaching a shorter course or wish to spend more time on some topic, the chapters on Beyond Sums of Squares and More on Prime Numbers are certainly optional in this sense.
The chapters concerning Points on Curves and Long-Term Function Behavior are not optional in my view of number theory, but may be viewed as โselected topicsโ.
The introductory (short) chapters 1 and 18 should not be considered optional, but may be emphasized or not to instructor taste. The point is just to motivate what we are doing before getting to formal definitions.
If you don't like cryptography or believe (like Hardy) that there are no applications to number theory, you can certainly create a nearly application-free course by skipping the chapters on An Introduction to Cryptography and Some Theory Behind Cryptography.
I don't consider the last several chapters on the prime counting function and other arithmetic functions connecting to calculus to be optional, but I have the luxury of having mostly juniors and seniors for a full semester. In a quarter course or one aimed more at sophomores (in the United States), one should still at the very least spend a couple days at the end of the course talking about these topics, perhaps discussing sections 21.2 and 21.3, and smatterings of Chapter 25.