Showing posts from May, 2016

Theta Functions

As mentioned in the previous post, modular forms are complicated mathematical structures that follow a number of symmetry rules. One of the typical modular forms is what's called "theta functions". A theta function TF over two variables a and b is defined as: TF(a,b) = Sum(-oo <= n <= oo) a^(n*(n+1))/2 * b^(n*(n-1))/2 Basically it is an infinite sum in both directions of a product whose variables are raised to n, which is the variable from minus infinite to positive infinite. The question is to observe the symmetrical properties of this form. One way to do that is by plotting the 3D chart for the following function: z = Sum(-oo <= n <= oo) x^(n*(n+1))/2 * y^(n*(n-1))/2 This was we transform the form into a simple z = F(x,y) function. Given a pair (x,y), we can then generate z and have the triplet (x,y,z). Let's write some code to do that. Code is straightforward but two simple aspects to be aware of: 1) If you do a for(int i=-oo;i<=oo;i++)

Breaking Fermat (Part 3 of 3)

How did Wiles prove FLT? I love that story, and there is a great book that recounts all that. Fermat claimed that he had a beautiful proof but never wrote it down. Most experts in the field today believe that he thought he had a proof, but it was probably incomplete, or maybe they are just underestimating Fermat's math ability. Before we get to Wiles', there is one guy who will play a major role in the final proof: Galois . This French mathematician came up with something called "Galois Representation" which will play a crucial role. Sadly Galois was having an affair with a hot woman who was married to the best sword duelist in town, and in a fateful duel Galois was killed, still very young. In the centuries that came after Fermat stated his conjecture, the proofs were going nowhere. People were proving it for "N=3". Then "N=7 and N=11". Then for "N of the form 2K+77" or things of that nature. Of course at this pace a final proof wo