What is Fermat's Last Theorem?

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Math? Really?

Don't worry, I'll talk about Star Trek too.

I'll start with some history. Pierre de Fermat was a French lawyer and a mathematician. I would like to say "born in ___," but as it turns out, there is some dispute on the topic. Fermat had a brother, whom their parents also named "Pierre," who unfortunately died quite young. This happened in either 1601 or 1607. The other year is the year our Pierre was born.

Pierre de Fermat was exceedingly clever, and quite a good mathematician. It was his work on geometric series that Newton poured over (in part) when he was trying to suss out that whole "calculus" thing. Fermat attracted a fair bit of (extremely nerdy) controversy in his career as a mathematician, particularly for his method of calculating minimums and maximums of functions. Descartes, of "I think therefore I am" fame had developed his own way of accomplishing this task, and particularly disliked Fermat.

Pierre de Fermat
Maybe Descartes had a point,
doesn't he look smug?

Fermat was a fan of a greek guy called Diophantus, and his book Arithmetica. Fermat owned a copy of the book, and in the margins he would write out theorems (a specific mathematical 'idea'), but he never included proofs. In fact, he would often be quite annoying on this front, describing his proofs as "miraculous," but not writing them them down due to "lack of space in the margins" (paraphrased). Then he died. These scribbled notes became known as "Fermat's Theorems," and throughout history, mathematicians have proven each and every one of them to be correct. Except one. This became Fermat's Last Theorem, a sort of Holy Grail for modern mathematicians.

Alright, there is one equation, but we'll get through it, I promise. Here it is:

What this equation says is that if you add two numbers, both raised to the same power, they can add up to another number also raised to that same power. An example would be using squared numbers: 3²+4²=5² because 9 + 16 = 25. In fact there are loads of solutions when you plug in squared numbers (n=2). Throughout history, no one has found a solution with anything higher than a squared number (no solutions for n>2). This was curious, because it asserted that this equation had an infinite number of solutions for n=2, but absolutely none for anything above n=2.

TL;DR: For this equation, nothing above n=2 is possible. No whole numbers will ever make the equation true. This is what Fermat claimed.

So mathematicians went to work trying to prove Fermat right once again. The business of actually writing down a mathematical proof is usually devilishly tricky (I'm told). You have to cover all your bases and wrap up all loose ends, so there are no flaws. Worst of all, you have to get everyone else to believe you.

At this point in the story, in steps Andrew Wiles. Andrew Wiles first heard about the theorem when he was about 10, and set out to prove the theorem. He soon learned that his knowledge was too limited, and he gave up proving the theorem, but still pursued a career in mathematics. A while later, he was presented with an opportunity to professionally try to solve the theorem by woking on another problem. He worked at it for years, and in 1994, published a proof for the Last Theorem. It didn't quite work the first time and he then published a final version a few years later that has stood up to all the scrutiny the mathematical community can muster.


So we've done it, we've solved the Last Theorem.


Star Trek. I did promise to talk about Star Trek in the beginning of this post. In an episode in second season of the Next Generation (Royale), the show opens with Picard thinking about the theorem when Riker steps in. He describes the theorem briefly and says:

"In our arrogance, we feel we are so advanced, and yet we cannot unravel a simple knot tied by a part-time French mathematician, working alone, without a computer."

This episode aired on March 27th, 1989. I enjoy thinking that a problem the writers of Star Trek thought would remain unsolved into the 24th century was solved within a decade of the writing of that episode. If one looks at the quote, we can perhaps feel a little bit arrogant about out mathematical prowess as a species, even if we've a long way to go on other fronts to catch up with Picard and his crew, it seems we are, at least, on our way.


Cheers, 

      - Scott



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