Theorem 11: the pigeonhole principle

The pigeonhole principle is a fairly simple idea to understand, and is extremely useful — mathematicians use it all the time.  Despite that, it seems not to be mentioned in most schools (in the UK).  This week, I’d like to tell you about it, and to give a nice application.  My understanding is that the pigeonhole principle was first stated by Dirichlet (although he didn’t call it that!), who used it to prove a result about Diophantine approximation — this is the application that I’d like to describe here.  The pigeonhole principle is sometimes called Dirichlet’s box principle.

Theorem 10: Lagrange’s theorem in group theory

The term has just started, and so I have been contemplating the exciting mathematics in store for the new first years.  This week I thought I’d tell you about one such result.

New students often find that some of the mathematics they meet at university is more abstract than the mathematics they studied at school.  Abstraction can be an extremely useful tool.  Broadly speaking, here’s what happens.  Mathematicians notice several examples of the same behaviour.  They want to explore why: what is it about those situations that leads to the same behaviour?  They try to write down a list of those key properties.  They then define a new object: it’s anything that has those properties.  (So all of the initial situations should be examples of this new kind of object.)  That should hopefully lead to a better understanding of what’s going on.  Moreover, if mathematicians can prove something about this object using only the knowledge that the object has those key properties, then they have proved a result about every example of that object — all in one go!

One good example of this is the notion of a group.  I’m not going to go through the definition of a group in great detail here, so if you haven’t come across the concept before then you might like to read this article on NRICH.  Here, I’ll remind you of the definition of a group and give a few examples.

Theorem 9: Bachet’s duplication formula

This week’s guest author is James Cooper.  Thanks, James!

Bachet’s Equation and Geometry

Today’s blog entry concentrates on Diophantine Equations — problems posed in terms of whole numbers, and connected problems over the rationals.  Typically these problems are very easy to understand but difficult to solve.  Their solution often involves leaving the safe world of the integers and using tools and techniques from other areas of mathematics before “projecting” the answer back into whole numbers.  The example I’m going to describe today will use ideas from algebra and geometry.

Fix some integer .  What are the rational solutions of the equation

?            

By rational solution, I allow and to be fractions, but not arbitrary reals.  So we are asking for the difference between a square and a cube to be a certain fixed integer.  This is known as Bachet‘s equation, and we will see that its Geometric interpretation is the key to generating solutions.

Radio programme

One of my favourite places for finding good mathematics in the media is In Our Time on Radio 4, a programme hosted by Melvin Bragg.  Each week, he picks a topic (from a very wide range of subjects) and discusses it with experts for forty-five minutes.  Properly.  Without avoiding the difficult bits, even when it’s maths.  Which I really like.  On the 24th September, he discussed the famous dispute between Newton and Leibniz about who came up with calculus first.  You can still listen to the programme online.  Not a huge amount of maths in this one, but jolly interesting nonetheless.  (You can probably listen to the other mathematical programmes in the archives too.)