M4P14 Elementary Number Theory Enhanced Coursework

This enhanced coursework is due in (to my office or my pigeonhole) on Wednesday, 21st January 2004. Early submission is encouraged.

You are supposed to write an essay on

Continuous Fractions.

Topics that you should strongly think about covering are the following:

• finite and infinite continuous fractions,

• convergents,

• rational and irrational continuous fractions,

• periodic continuous fractions and quadratic irrationals,

• applications to approximating real numbers by rationals.

Then you can go on and explore the method of constructing solutions to Pell’s equation n^{2} − am^{2} = 1 using continuous fractions. Alternatively, you may want to explain how to write the continuous fraction for the number e. There are other interesting things that you may like to explore (e.g. geometric interpretation of continuous fractions).

Some references: Davenport “The higher arithmetic”, Hardy and Wright “An introduction to the theory of numbers”, LeVeque “Topics in number theory”, Lang “Introduction to Diophantine approximations”.

But look at other books as well! It will help make your project personal, which is always nice, and is likely to attract more marks. If I see something that you have done and that no-one else has, this will make me feel very positive about your project.

Please do not copy things straight out of books or off the internet. You won’t get many marks for this. It is very important to convince me that you understand what you are writing about.

I expect the project to be 15 to 20 typed pages. Handwritten projects are accepted.

Alexei Skorobogatov, 3/11/2003