# Fermat's Last Theorem

In mathematics, guesses are called conjectures. Once a conjecture is proven, it becomes a theorem. This is a major part of what being a mathematician is all about: proving conjectures or, if we were to wax lyrical, seeking truth.

**1637, France.**

A lawyer was reading an ancient math text, and he came across an equation - an equation that some of us might remember from school: the famous Pythagorean theorem.

However, this seemingly innocent-looking mathematical statement gave our man a pause, and he wondered. We don’t know for how long – days, weeks, months – perhaps, or maybe it was all obvious to him in an instant (unlikely, but the point is we don’t know). What we do know is that in the same text he was reading, he eventually scribbled down another equation, one which looked awfully similar yet, in some sense, profoundly different. And not just that; against that equation, he famously went on to scrawl, “I have discovered a truly remarkable proof for this proposition (equation), but this margin is too narrow to contain it.”

In fact, in the same text, he had scribbled down many other guesses, conjectures. What is surprising is that over the course of the next few decades and centuries, every single one of those guesses would be proven true except one – if only the margins weren’t as narrow!

The lawyer in question was one of the greatest mathematical minds of his generation. Although not a mathematician by profession, he was nonetheless quite the craftsman. Pierre De Fermat loved the subject and pursued it in his spare time. The conjecture no one seemed able to prove came to be known as Fermat’s last theorem."

For the next 350-odd years since it was first published, people tried and failed. A legend grew, and the attempts to prove this ever-so-elusive conjecture led to exciting new discoveries in maths. However, the eventual proof continued to elude everyone. No one knew for sure if it could ever be proved, or whether it was a conjecture even worth pursuing!

But that didn't stop people from believing. In fact, quite a few of them hoped against hope that perhaps, after all, it could be proved. One reason for their hope was most likely its seemingly simpler appearance, which made this unproven conjecture ever so enticing to both amateurs and experts alike. Fermat's last theorem merely states:

for any integer value ofn > 2.A smart 10-year-old could read and understand this equation.

For all its allure, Fermat’s last theorem would go on to gain the dubious distinction of being the equation with the greatest number of successfully published unsuccessful proofs**.**1 2But things would change, as they always do.

Come 1993, a mathematician at Princeton, Andrew Wiles went on to tell the world he had the proof. A media frenzy ensued, and he was covered by the Time, CNN and who have you! Overnight, he became somewhat of a global sensation.

He eventually delivered his proof in a three-part lecture series at Cambridge, and caused the world to rejoice. Finally, the monster was slain; the equation had been conquered. It was time for a victory lap.

Unfortunately, that victory was short-lived. They discovered a flaw in his proof, and Wiles was forced to go back to the drawing board. Before announcing it to the world, he had been working on the problem in secrecy for the past 6 years but had been fascinated by it for the past 30, ever since he was 10 years old. However, this time around, it was different. It was no longer a secret; the world had expectations. Just a couple of months ago, he had been deemed one the greatest ever, but now with a mistake so public, he was humiliated.

Nevertheless, he went back to try and fix it. At first, the problem seemed minor, fixable even. However, the more Wiles tried, the more he couldn’t resolve it, and another year passed.

He was on the verge of giving up and had decided to share whatever he had worked on so far with the world because he clearly felt he was not up to the task.

On September 19th, 1994, a little over a year after the famous mistake**,** Wiles recalls *“I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day, I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself; I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.”*3

And that, ladies and gentleman, is how it was done!!