html>Number Theory

Ma458 Number Theory

Ethan Bolker
Fall 2013


Math majors, both those planning on graduate school and those planning to teach high school. Computer Science majors looking for a theoretical elective. High school math teachers. Anyone who likes mathematics and wants to learn more.

Goals of This Course

To learn some number theory, of course. To learn why you might want to learn some number theory (if you don't already know). To enjoy learning number theory. To practice reading and writing mathematics. To stretch yourself intellectually.

Course structure, grades

I expect you to pay attention in class, to do the homework, to read about number theory in different places and to ask questions about things you do not understand.

There will be lots of homework. If this is your first course beyond calculus and linear algebra you may find the problems different from those you have encountered before in a math course. Few of them have what one could narrowly call an answer. All require playing with numbers, observing patterns, writing convincing arguments.

There will be a midterm and a term paper. The final examination will probably be (mostly) take-home.

I am considering structuring a significant fraction of the semester as a seminar. If I do, each student will present. All will be responsible for all the material covered.

I do not use an algorithm to compute grades. Here's roughly how I weigh the factors:

I know that the students in the class are starting from different places mathematically. I expect each of you to learn at a level that's appropriate. So the same score on an exam may mean an A for someone and a B for someone else.

I do not grade on a curve. Everyone can earn an A. (Wouldn't that be a wonderful class!)


Course meetings:

TTh 11-12:15, Wheatley W02-0124

Getting help and encouragement

I am always glad to talk about mathematics. Whenever my door is open (most of TTh) you are free to ask questions. Email to eb at cs dot umb dot edu is the best way to reach me, both during the day and evenings and weekends. My office phone is 287-6444 but I don't have voice mail and I.m not always there. My home phone is (617) 969-2892, but please use that only for real emergencies. (Emergencies that happen after 8 PM should wait for the next day.)

Often other students in the class can help you. It's worth learning how to work with, learn from (and help) your peers - how else will you keep learning when you've left school?


The text for this course is Joseph Silverman, A Friendly Introduction to Number Theory. The first six chapters are available on line at that site, which will give you time to get the book.

Here are several others you may find might find useful. The first few are inexpensive and I recommend them highly.

Use of Computers

None is required. Some of the topics we touch on have applications in computer science, and we may mention those applications, particularly when we learn about cryptography. But number theory is an old discipline, and this is an old fashioned course. No technology. All you need is pencil and paper - perhaps a calculator if you're not fast at arithmetic. But there are infinitely many integers and almost all of them are larger than your calculator can handle, so you can't substitute calculation for understanding.

That said, you may find that automating some computations helps you understand the number theory, both because it gives you access to larger examples than you can construct by hand, and because to code an algorithm you must master it. In the past students have written programs in C, Pascal, Scheme, mathematica and for their programmable calculators.

If you want to use a computer and don't want to write your own programs, there is lots of number theory software out there. Most of it is for professionals but there are packages that are essentially instructional. I don't know of any, but would be pleased to help you decide whether one you found was worth working with.


Section 504 of the Americans with Disabilities Act of 1990 offers guidelines for curriculum modifications and adaptations for students with documented disabilities. If applicable, students may obtain adaptation recommendations from the Ross Center for Disability Services, M-1-401, (617-287-7430). The student must present these recommendations and discuss them with each professor within a reasonable period, preferably by the end of Drop/Add period.

Student Conduct

Students are required to adhere to the University Policy on Academic Standards and Cheating, to the University Statement on Plagiarism and the Documentation of Written Work, and to the Code of Student Conduct as delineated in the catalog of Undergraduate Programs, pp. 44-45, and 48-52. The Code is available online at

For my own elaboration on the rules concerning the acknowledgment of intellectual debt, particularly appropriate in this course where the focus is on teamwork, see