## Sutton Trust summer school 2012

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 $\sqrt{2}$ is irrational, and talked about continued fractions.  There are many proofs that $\pi$ 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.