As all stories should, let's start with Pythagoras. It was a sunny day in Ancient Greece...
Nah, on second thoughts, let's not. Big Bang Theory already took that one. Anyway, a long time ago, in a country far, far...
Hang on. No, Star Wars had that...
Anyway. Greece. Home of Heracles and hummus. And someone who may or may not have been a dude called Pythagoras, in his almighty (possibly bearded) wisdom, decided that right-angled triangles should be easier to understand. With perhaps the most well known mathematical formula of all time: a squared + b squared = c squared. (Blogger doesn't allow for superscripts, so sentences will have to do, get used to it)
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| You all remember this from GCSE, right? You will be tested on it. |
Of course, there's no reason whatsoever to stick to just whole numbers... unless you happen to be a number theorist, a strange bunch of mathematicians that spend their time obsessing over whole numbers. If it ain't an integer, they don't wanna know.
But let's get back to the story at hand. Pythagoras (or maybe some other Greek dude/girl, no-one's quite sure really) devised this triangle thing, and everyone was happy for about 200 years. Then a guy called Diophantus, who is much less known outside of the world of mathematics, wrote a book called Arithmetica. While writing up Pythagoras' theorem, Diophantus (or maybe someone earlier than him, seriously this happened thousands of years ago, stop expecting perfect historical accuracy) said to himself, "Well, why do we have to stick with using squared? What about cubed, or to the power of four? What about ANY power larger than two? Are there any numbers that work like that?"
No-one answered him. Mostly because he was probably sat alone in a room writing his book and no-one heard him, but still. Taking a few weeks off from writing, he had a go and couldn't think of any solutions to this equation, written out as a^n + b^n = c^n for some n>2. So, in his book, he wrote that this equation had no whole number solutions.
The problem? He couldn't explain why. And saying you can't do something and giving up isn't proof. It's like saying humans can't survive without air and then NOT strangling a person to prove it? Maybe? I dunno. And so, for a thousand years, mathematicians tried to figure out a proof for this problem. And quite frankly, checking every combination of integers to see if one works takes a little too long. As in, if you'd started counting at the dawn of time, you might have made it about 0% of the way through by now. Cos there's a looooot of integers, and a looooot of possible n's to check them for.
So, people sort of got bored, called it impossible to prove, and moved on. Until, in 17th Century France, a dude called Pierre de Fermat (who we know a lot about due to it being so recent. And no, he definitely didn't have a beard. At least, not in his Wikipedia portrait), who was a lawyer by day and a number-crunching, crime-fighting, maths wizard by night (except he probably fought crime as a lawyer too, so that's what he did from lunch to early evening), was flicking through his own copy of Arithmetica when he spotted this equation. Having a stab at it, he did some quick mental arithmetic, came to a conclusion, and wrote on the edge of the book, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
And then he died.
Wonderful.
Well, it was 30 years later that he died, but he never actually wrote down the proof that he claimed to have found. After the funeral, which probably consisted of a quiet ceremony and a much less quiet reception, Fermat's son Clement-Samuel, came across his father's notes, and subsequently republished them as that year's Christmas best-seller. Seriously, this was huge in the maths community (which is small, secluded and somewhere in Berkshire), and every number theorist in the world tried to find Fermat's solution as their own.
And... no-one did. A lot of people said that we just couldn't understand the maths that Fermat had thought up. A few said that Fermat must have been mistaken. And a couple just thought he was leaving a huge trollface in the middle of his textbook. Who knows?
Fast forward another 300 or so years. It's the 1960's. By now, people have devised ways to prove the theorem in bits and pieces, but no general solution exists. And in steps a 10 year old kid called Andrew Wiles. Reading a textbook on his way home from school, Wiles comes across Fermat's Last Theorem and, as all good mathematicians do, becomes obsessed with solving it. Showing it to his teachers, family and even a guy in the street called Gerald, none of them could explain to him how a theorem so simple to describe hadn't yet been proved. Except Gerald, who, other than being a mathematical genius and world-renown gymnast, was most likely made up. By us. Just now.
As most ten year olds do, Wiles grew up, went to University and attained a Ph.D. in mathematics (although a lot of 10 year olds get their degrees in other subjects). Starting a research grant in the mid-80's, he and a group of number theorists used a lot of time, effort and complicated maths to figure out a proof. But after a year, they hadn't made much progress. Despite this, Wiles continued with his work alone, in almost complete secrecy (he told his wife and that was it), continuing for 6 years to find this general proof.
And he got one. In 1993, Wiles published a paper which definitively proved that Fermat's Last Theorem was, in fact, true.
And then in August 1993, someone spotted the error he'd made.
Oops.
Wiles was, as is to be expected, a little discouraged by this. 7 years of his own work had been based on a flawed bit of maths. Disappointed is a bit of an understatement for this.
But, being British (and a mathematician), Wiles didn't give up. Devoting his entire researching timetable to the problem he worked for over a year to rectify his error. In time, he came to realise that the error, although incorrect, was a crucial part of his argument. He couldn't simply remove the error and call it a day. It seemed that the end had come.
Until, in 1994, on the verge of defeat and looking at his working one last time, he noticed a way around. A way to avoid the error entirely, which was both simple and valid. Writing this solution, Wiles was finally able to call it the finished article. His published findings were scrutinised, verified and accepted by the mathematics community. Finally, Fermat's Last Theorem, first proposed by Diophantus two thousand years previously, had a full working proof.
*fireworks, cheering crowds, etcetera*
So there you have it. A very short summary of one of the most interesting problems in all of mathematics. So simple to state even a 10 year old can understand it, but so difficult to solve that the greatest mathematicians of the age for two thousand years were stumped. Goes to show that all it takes to solve a several millennia old problem is one British dude in his mid forties. Words to live by, right?
Posted by Jake, but Baker did most of the work. :P
