Welcome to the course blog for the Cambridge Part II Number Theory course (Michaelmas Term 2011). I hope that this will be a useful resource to accompany the lectures, examples sheets and supervisions. Please check back regularly.
The plan is that I’ll put up a new post just after each lecture. Each post will have a quick summary of the topics covered in that lecture, with suggestions for further reading, plus one or more problems to get you thinking about the topics that will be covered in the next lecture. Please do try these problems, as they will help you to get the most from the lectures (even if you don’t solve a problem, thinking about it will still be useful).
You are very welcome to leave comments (for me and for your fellow students) on each post. For example, you might have your own suggestions for good places to read about the topics, or you might have another way to look at one of the ideas, or you might have a really good example that illustrates some interesting aspect of the material, or you might have a question that you’d like to raise. I also encourage you to post to let me and others know how you and the people you’re working with have got on with the problems for the next lecture; please feel free to share ideas here (they don’t have to be complete solutions).
Of course, now I have to suggest something to think about before Lecture 1! So here it is.
In the first couple of lectures, I’ll go over some topics that will hopefully be familiar to you from IA Numbers & Sets: divisors, prime numbers, Euclid’s algorithm, Bézout’s lemma, the fact that if a prime divides a product then it divides at least one of the factors, and the Fundamental Theorem of Arithmetic, for starters. I’d like you to think about how you would present this material to someone who hadn’t seen it before. (Better still, think about it and then find a IA student who doesn’t know the material, and try to help him/her to understand it!) How will you describe/state the results? What examples will you choose to illustrate the ideas? How will you prove the results? How will you order the material? And how will you motivate the ideas (that is, present them in such a way that they seem natural)? If you can get the ideas clear enough in your own mind that you can convey them to someone else, then you will have a pretty good understanding.
And here’s a picture to get you thinking. What might it illustrate? Can you produce an example of your own? (You can click on the picture to get a larger version.)