 I have a hilarious joke about Fermat's last theorem. Unfortunately, it's a little too long to include here. Have you ever heard a fairy tale about mathematics? Check this out. Pierre de Fermat was a wealthy magistrate in 17th century France. Like many other wealthy individuals around that period, he cultivated several hobbies to help him pass the time. And remember, video games weren't an option yet. He was a successful poet and he was fluent in several languages. And some evenings, he enjoyed cracking open a bottle of Bordeaux and doing a little math. He also happened to have one of the finest mathematical minds humanity has ever produced. You know how there's a ton of stuff named after Descartes and Fermi? Well, same thing for Fermat. His work was instrumental in helping Newton and Leibniz invent calculus, and he made huge advances in probability theory, number theory, analytic geometry, all sorts of stuff that's hugely important to the field of mathematics to this day. Like I said, video games weren't a thing yet. Because it was just a hobby, Fermat wasn't really forthcoming about his work. He never published anything himself. He just made offhand references to his revolutionary ideas and letters to friends and in the margins of books he was reading. For him, it wasn't really about recognition or improving the field. It was just about the thrill of having figured something out. He actually said, I have so little aptitude in writing out my mathematical demonstrations that I have been content to have discovered the truth and to have the means of proving it when I have reason to do so. Basically, so long as I know, who cares? As such, he rarely wrote out his ideas in full. Instead, he'd have some scribble or another that referenced some incredible mathematical insight, with no explanation whatsoever about how he'd figured it out. One such comment was made in the margin of an ancient Greek text called Arithmetica. You probably know the Pythagorean theorem. For any right triangle with legs of length a and b, in hypotenuse c, a squared plus b squared equals c squared. There are an infinite number of combinations of a, b, and c for which this equation is true, one for every single right triangle you could draw. But what if we screwed with it a little bit, just for kicks? What if we restricted ourselves to integer values that weren't zero, so no fractions or decimals? And what if we bumped up each one of those exponents by one? So now it's a cubed plus b cubed equals c cubed. What three whole numbers could make that work? I mean, this is elementary school math. Just about anyone could pick up a calculator and find an a, b, and a c that could do this, right? Feel free to pause the video and try it yourself if you want, but I'll warn you now, it's not easy. It's not easy because there aren't any. The same goes for these three numbers to the fourth power, or the fiftieth power. In fact, for any exponent greater than two, there are no three non-zero integers that satisfy this equation. Fermat wrote this observation down in the margin of Arithmetica, saying, I have a truly remarkable proof which this margin is too small to contain, and then he never told anybody about it. In the years following his death, after his son made all of his notes public, mathematicians poured over all of these little scribbles and discovered that for the most part, the ideas that Fermat never really fleshed out in writing were mostly right on, that all of his little conjectures and observations usually had very clever and insightful proofs, except this one. I mean, they really tied themselves in knots over this thing, and continued to do so for years and years after all of his other theorems had been either proven or disproven. It seems so absurdly simple. Either prove that it's right, or find three numbers that work here and prove it wrong. And yet, this last remaining theorem sat around for centuries. Trying to figure out what the hell Fermat was thinking led to all sorts of other discoveries, but even with all new branches of mathematics and mathematical tools, Fermat's last theorem just wouldn't budge. Fast forward to around the mid-90s. In 1995, some of those new mathematical tools were used to prove something called the Taniyama Shimura Conjecture, or the Modularity Theorem. The proof is an astonishing 150 pages long, and it was compiled by a guy named Andrew Wiles. Wiles is a professor of mathematics at Princeton University. He's been a fan of math ever since the age of 10, when he learned about Fermat's last theorem in a book, and dreamed of being the one to figure it out. Of course, that was just a childhood dream, but it really drove him to learn and get very good at advanced mathematics, the sort of stuff that you have to be very good at to even begin to think of proving something as beastly as the modularity theorem. There's a reason the proof is 150 pages long. It involves elliptic curves and morphisms and number fields and parameterization and all sorts of crazy stuff. Most mathematicians thought that it was way too complicated to actually prove that it might be an interesting observation, but we really needed to invent some better mathematical tools before we could really pin it down. But interestingly enough, it was known that the modularity theorem contained Fermat's last theorem inside it as a special case. If you were able to prove it true, then you would necessarily have also proven Fermat's last theorem true, and Wiles was a pretty audacious dude with a childhood dream. After working in secret for seven years, Wiles finally released his proof to an astonished world, and it wasn't quite right. Another mathematician had found an error in his proof. One error in 150 pages of some of the most advanced mathematics to date. I screw up calculating the tips sometimes. Everyone held their breath while he took another couple years to puzzle over the error, and then, with the help of one of his students, he realized that there was a way around it, an elegant shortcut that he hadn't thought of, and he fixed it. Finally, Fermat's last theorem had been proven. Sir Andrew Wiles was knighted for his heroic effort, for having the courage to work in secret for the better part of a decade on a problem that had stymied some of humanity's greatest brains for hundreds of years, and to have eventually persevered against it. But, despite that triumph, there's still a whisper of uncertainty. We don't know what Fermat was thinking when he scribbled this note, but the fields of mathematics that Sir Wiles used to slay the dragon of the modularity theorem were only invented relatively recently. Fermat wouldn't have had access to any of that. He was still working with relatively simple 17th century stuff, which is a problem. I mean, it's possible that he just made an error and thought that he'd discovered a simpler proof. There have been several mathematicians who got something that looked kind of right, but fell apart if you look closely enough. But it's also possible that he thought of something incredibly simple and elegant, something that didn't take 150 pages of 21st century mathematical brilliance to manage, but just a very clever 17th century judge who didn't have video games. A proof has been found, but some mathematicians are still searching for something short and ingenious, for something subtle and brilliant, for something that Fermat himself might have come up with. I imagine that if such a proof exists, step one is going to be, pour yourself a glass of wine. Do you think Fermat's last theorem has a shorter proof we have yet to find, or do you think that Sir Wiles was the first human to discover it? Please leave a comment below and let me know what you think. Thank you very much for watching. Don't forget to ball ball subscribe, blush air, and don't stop thunking.