Armed with highlighters, pens, and a resolve to read, to learn, to understand, I printed out all 109 pages of Wiles’s proof of Fermat's Last Theorem, minus the references, anticipating a challenge like A Brief History of Time. In fact, I used the exact same system as I did whilst reading Hawking’s work: purple highlighter for things I don’t understand, yellow for things and I do, and a crapton of pens for trying to work it out in the margins. There’s something vaguely poetic about that, to me, considering A Brief History of Time started the chain of events which led to my capstone topic choice, and maybe I unconsciously hoped to mirror that unlocking of understanding and curiosity by reading Wiles’s Proof.I fear there aren’t enough purple highlighters in this world to fully demonstrate how much I did not understand the proof for Fermat’s Last Theorem. 

Somehow, I still clung to a shred of hope that something had worked, and I looked back through the pages and pages of what was supposed to be one of the greatest mathematical proofs in all of history, but it  had somehow turned into the coloring pages of that one kid in my kindergarten class who really liked purple. Wiles, whose target audience was the top mathematicians in the world, jumped straight into Galois Representations and different fields and rings and points at infinity and deconstructions and, most importantly, modular forms. In an instant, I realized why I was so completely and utterly lost: Wiles’s target audience wasn’t some high school student who hadn’t even taken linear algebra. I could decipher Hawking because he was writing for those who didn’t know anything, but Wiles assumed a base of at least graduate school in mathematics. To twist the knife already embedded in my pride, I remembered the words of Wiles’s friends in the NOVA documentary, who were all asked to describe modular forms for the average viewers. Cut to a shot of each and every one of them laughing. That moment marked when I realized just how ludicrous it was for me to even try to look at this proof.

However, I emerged from this move with two great victories: how Fermat's Last Theorem connects to modular forms, and a basic understanding of cryptography. Let's begin with the former.

In 1984, Gerhard Frey rewrote the Fermat’s Last Theorem equation as this:

This looks clunky and vaguely unuseful, but for a minute consider that

Now, the equation looks like this:

Now we can appreciate its elliptic nature. I stated just before how clunky the equation appears, but that’s a positive sign; the E series of this equation has a very unique set of numbers, such that the probability it has a matching M series is essentially inconceivable. Based on Taniyama-Shimura, if it has no matching M series, this equation does not exist, as every elliptic curve is modular. And if this equation does not exist, it yields no real answers, proving Fermat’s Last Theorem to be correct! This is the connection between the Taniyama-Shimura conjecture of the elliptic/modular words and Fermat’s problem of geometry. The way this functions in terms of mathematical reasoning is in a proof by contradiction. It's displayed to the right.

The result, of course, is that if the Taniyama-Shimura Conjecture is correct, Fermat's Last Theorem is also correct. I have no understanding as to how Wiles proved Taniyama-Shimura, but I was interested in elliptic curves, which lead me in turn to cryptograhy. I learned how to write RSA and elliptic curve encryptions, but because they function similarly, I'll only detail the RSA encryption here.

HOW TO WRITE A USEFUL ENCRYPTION

RSA algorithms are trapdoor functions: they depend on the idea that it’s easy to go through the door one way, but difficult the other way. The algorithm does this by using the concepts of multiplication and factorization, as it’s quick and simple to multiply two numbers, but time-intensive to take a number and break it down into its prime components. One would assume, in this case, that the product of the two numbers should be arbitrarily large, right? Wrong. Computers don’t do well with large numbers, so in order to account for this, we set a maximum (max) by multiplying two random prime numbers. In order to best explain the concept, we’ll use a dummy example: the word CRASH.

If we run a decryption, we receive 'CRASH'.