(Picture from the Simpsons episode “Treehouse of Horror VI” where Homer enters the world of 3D.
Try entering the numbers in your hand calculator)
The statement of the perhaps most famous mathematics problem over the past few centuries should not have slipped past many people with some interest in mathematics.
I am of course talking about Fermat’s Last Theorem (FLT) which was finally resolved by Andrew Wiles in 1995 in what is considered one of the most remarkable mathematical achievements of modern day.
FLT is one of those problems whose statement is so often understood by anybody on the street, but is very difficult to prove. The amount of machinery that lies behind Wiles proof is astonishing (more so in algebraic geometry than pure number theory).
As a reminder I will state FLT and then begin discussing the contents of this post.
In this post I wanted to treat a related question.
Is FLT true over finite fields of characteristic p?
That is, can we refute the existence of non-trivial integer solutions to if we work where is a prime number?
The answer is No.
You can probably immediately spot counterexamples such as .
More remarkably however, given there exists non-trivial solutions to for every prime sufficiently large.
Schur came up with this result while trying to prove FLT.
This result probably gives some idea as to why FLT was so hard to prove since it shows one cannot simply consider congruences like one often does when working with other types of Diophantine equations.
Another interesting fact about this result is that it can be proved using both Fourier Analysis and Ramsey Theory. This may seem rather surprising since Ramsey Theory is combinatorics (a field dealing with countable discrete structures) whereas Fourier Analysis revolves around function approximation using continuous trigonometric functions, and both are used here to prove a number theoretic result. This shows just how differently mathematics problems can be approached if one can find a suitable phrasing in whichever language is being used.
I would like to focus on the latter proof using Ramsey Theory.
So what is Ramsey Theory?
The slogan for Ramsey Theory is basically that any sufficiently large structure, no matter how disordered, contains some element of order. For graph structures one usually likes to look at edge colourings and say something about specific subgraphs where all edges are of the same colour.
Let’s make an important definition to make this more precise:
I have omitted one crucial detail: How do we know is well-defined?
This is exactly what Ramsey’s Theorem asserts, that this number exists in the first place.
The proof is not actually very difficult, one proceeds by proving an upper bound on . One can then show well-definedness of the generalized Ramsey number by proving the recursive inequality and using the well-definedness of as part of an inductive argument.
I will however omit the details of the proof and move on to the main result of this post.
Before we get there we need an intermediate theorem which seemingly has little to do with our goal: