We had three lectures on `Numbers’, as part of the Cambridge Maths Sutton Trust summer school in August 2012. 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. We mentioned that there are many more irrational numbers than rational; that’s because the rationals are `countable’, whereas the irrationals are `uncountable’. A couple of the problems on the examples sheet are related to the Chinese Remainder Theorem.
Other mathematicians who were mentioned during the week: Cantor, Cardano, Diophantus, Fermat, Galois, Tartaglia, Wiles. At some point we mentioned that there are infinitely many primes. We saw a special case of Fermat’s Little Theorem.
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 have any questions about the maths arising, or have suggestions for good places to read more about the material that you’d like to share, please do leave a comment below!