The official prerequisites for this course are Math 301 or Math 395. However, not much of the material in either of these courses will be used. What is important is that you are used to thinking analytically and (at times) abstractly. Since this is a rigorous, proof-oriented course, it would be helpful if you have taken such a course before, but this is certainly not necessary.

There will be approximately 10 homework sets throughout the semester. Your total score on these sets will be prorated to a score out of 200 when I calculate your final grade.

The majority of the homework problems in this course will be theoretical in nature. That is, you will need to prove something rather than just writing down a numerical answer. When doing these problems, it is important that your final answer be well-written. In other words, you should make your arguments using full sentences, and sometimes even paragraphs. Of course, there will usually be equations or other mathematical symbols too, but they should be woven into the paragraph, rather than just on the page all by themselves.

If this sounds hard, then I'll admit that it can be very difficult at first. However, my experience is that most people figure out how to write mathematics fairly quickly. Also, when I prove theorems in class, I will usually write out complete proofs, and will always tell you when I'm just outlining how a proof should go without giving you a complete proof. So you will have lots of examples to look at.

While most of the problems will be theoretical, there will also be some that are strictly computational in nature. For those problems, it is important that you do not simply write the answer - to get full credit, you need to show the work you did to execute the computation.

When doing homework, you are allowed (and even encouraged) to work on problems together. However, if you do this, please be aware of the following rules. First, working together means that everyone works together on a single problem. It does not mean that James does the first problem and Stacy does the second. Second, if you work with someone else on a problem, tell me at the beginning of the problem who you worked with. (I am serious about this! When you work with somebody else, it is important to give them their proper credit.) Finally, after you discuss the problem and figure out how the proof should go, each student should write up the proof separately in his or her own words. If you find some of the homework problems to be extremely hard, or you're having trouble figuring out how to write up a solution, feel free to come see me in my office hours. If you're collaborating with friends, feel free to all come together as a group.

Number theory is the branch of mathematics which studies properties of numbers and how they relate to each other. For example, one theorem which we'll prove this semester is that any positive integer can be written as a product of prime numbers in an essentially unique way. (When I say "unique", I mean that we'll consider the products 3*5*5,  5*5*3, and 5*3*5 to be the same.) Another theorem, which we will not prove in class, is that any positive integer can be written as the sum of at most 4 perfect squares (for example, 231 = 15² + 2² + 1² + 1²). Actually, this is a rather naive definition of number theory, since there are many other topics that also fall under the heading of number theory. But this is a good starting point.

On the syllabus page, you can see essentially the titles of each section. Here I'll try to give a bit more of an overview of the content chapter by chapter.

Chapter 1: Sections 1-2.

In this chapter, we'll first learn about mathematical induction, which is a powerful technique for proving things in mathematics. Then we'll use this technique to show that that any integer has a unique representation in any base. For example, we usually write numbers in base 10.

Chapter 2: Sections 1-4.

In this chapter, our main goal is to prove the fundamental theorem of arithmetic - than any positive integer can be factored into primes in an essentially unique way. Along the way, we'll talk about basic divisibility properties of numbers, greatest common divisors, and how to find all integer solutions (x,y) of equations such as 9x+5y=2.

Chapter 4: Sections 1-2.

In this chapter, we'll begin to study congruences. The most familiar example of congruences is congruence modulo 12, which is the same as adding numbers on a clock. For example, if it is 10:00, and you want to know what time it will be in 4 hours, you count around the clock to your answer - 11:00, 12:00, 1:00, 2:00. So your answer is 2:00, and not something like 14:00. Basically, what you're doing is starting over whenever you get to 12:00. In this chapter, we will make this idea mathematically rigorous.

Chapter 5: Sections 1-4.

In this chapter, we'll study how to solve simple (linear) equations involving congruences, and also study some other important properties of congruences.

Chapter 7: Sections 1-2.

In this chapter, we'll study primitive roots, which are another important topic in congruences. They are a bit too complicated to explain right now, so I'll wait until we've studied congruences a bit before trying.

Chapter 9: Sections 1-4.

In this chapter, we'll start talking about quadratic residues. These are the elements in a congruence which are perfect squares. Sometimes this will be obvious (the number 4 is a perfect square in any congruence because the integer 4 is a perfect square). However, sometimes it will be harder (if we look at congruences modulo 13, we will learn that -1 is a perfect square, even though the integer -1 is not). We will study a way to test whether a particular integer is a perfect square modulo a given number, and will also study the law of quadratic reciprocity, which is a very important theorem in number theory.

Chapter 10: Sections 1-2.

In this chapter, we'll study some of the deeper properties of quadratic residues.

Chapter 6: Sections 1-4.

If we have time, we'll go back and study the Euler phi-function, which is an important function in number theory with some very interesting properties. We'll also study the mu-function, which is also important and has interesting properties.

Chapter 8: Sections 1-2

In this chapter, we will study properties of the function pi(x), which tells the number of prime numbers less than or equal to x. For example, pi(10)=4 because there are 4 prime numbers less than or equal to 10. Similarly, pi(13)=6. The big thing that we will prove here is Tchebychev's Theorem, which gives upper and lower bounds for pi(x).