I have been reading Elements of Number Theory by John Stillwell for the last three years and I have only just started Chapter 8 (out of 12). It was a birthday present and so I would say it has been good value for money! Not that I have been reading it consistently. Sometimes I don't seem to have time to read much for several months and then I will pick it up again and have a burst of working and reading. The way I like to work through a text like this is to read and understand all the proofs and to work religiously through the exercises. Sometimes the proofs require a bit of extra work on paper to be able to understand them. Other times, by thinking about each sentence, I can understand a proof without the need to write anything down.
It may seem a bit bizarre to work through all of the exercises in the order in which they are presented but I find that if you complete one it gives you the confidence to tackle the next one. It also helps with the understanding of the text and, if there weren't exercises, all those proofs would seem a bit dry. So far I have done them all apart from those in section 3.8 until the end of chapter 3. I stopped working on this chapter when I gave up my OU studies as I felt I needed to start afresh on a new chapter. I have covered 56 pages of A4 with my workings! It would be nice to publish them all here but I don't think the publishers of the book, or the universities that use it, would be very happy about this.
I have finished chapter 7 on quadratic integers. This culminated in a proof of Fermat's last theorem for n=3 which I was pleased as punch to understand, especially when it says that this is "probably the most difficult result in this book...". It looks horrible when you just read through it quickly and nothing seems to make sense. You just have to take one sentence at a time and it may be that to understand a small point you have to go away and work at it until the penny drops. This was certainly the case for understanding the congruence classes mod √-3 in Z[ΞΆ3] in figure 7.3. I had to go back to basics to get to grip with this but it was key to some of the pre-stage proofs.
The next chapter in the book is about the four square theorem; that every natural number is the sum of four squares. For this we are introduced to Quaternions and Hurwitz integers. It all looks very intriguing.
Anyway, for all those of you who will be medling with some maths this Christmas instead of the usual crosswords or suduko, have a great time. Happy Christmas!
