We had three lectures on `Numbers’, as part of the Cambridge Maths Sutton Trust summer school in August 2013. Here are some suggestions for further reading.
In the first lecture, we talked about Euclid‘s algorithm and Bézout‘s lemma. In the second, we mentioned the Fundamental Theorem of Arithmetic, and we learned about modular arithmetic. In the third and final lecture, we mentioned that is irrational, and talked about continued fractions. There are many proofs that is irrational. A couple of the problems on the examples sheet are related to the Chinese Remainder Theorem.
Other mathematicians who were mentioned during the week: Diophantus, Euler, Fermat, Germain, Riemann, Wiles. At some point we mentioned that there are infinitely many primes. We talked a little about the Twin prime conjecture, and about recent progress towards it; we mentioned that there is a recent article on Plus about this.
Any book that is an introduction to number theory will cover these topics. My favourite (of the ones I’ve read) is The Higher Arithmetic, by Davenport (published by Cambridge University Press) — it has loads of great maths, presented in a way that I find very readable.
If you’re looking for good books to read in preparation for a maths degree, then you might like to look at this page that I wrote for the Murray Edwards College website (that’s the college where I’m Director of Studies). For other assorted interesting mathematics-related things, you could watch the Maths at Murray Edwards Facebook page.
If you have any questions about the maths in the lectures, or have suggestions for good places to read more about the material that you’d like to share, please do leave a comment below!