So, today I'm going to tell you about
what's called Catalan's conjecture. So even though it's called a conjecture,
it's actually known, so this was proved by a mathematician
about 15 years ago. And it's a question about,
or a statement about, perfect powers, okay? So... What do I mean
by perfect powers, first of all? I mean, take some whole number -
1, 2, 3, so on - and raise it to a power which is larger than 1, okay, so squares, cubes and so on. So if you start to write these things down, 1 is a perfect power, in fact a power of itself to any exponent, 2 is not, 3 is not, But then we get 4, which is 2 squared. Then we have to go up to I guess 8, which is 2 cubed, and 9 is 3 squared, and 16, 25 Am I gonna run into any cube soon? Let's see Oh 27 and 36 and 49 and so on - so these are numbers that can be built by taking two integers and raising one to the power of another? -that's exactly right Yeah, so of course all we've seen so far mostly squares and cubes I guess 16 is also a fourth power But you can take the exponent to be as large as you want, raise it as high of a power as you want, and you Can take the number to be as large as you want to. I'm just trying to put them in order. -so do these numbers become More common or less common as we go down the number line? -so they become less common and generally you think they're spread out right, like just think about squares for example if you hand me a big number like a million and you want to know "about how many squares are less than a million?" well, It's about square root of a million right because that's how small the number has to be To be less than a million after you square it. they're cool numbers They're kind of spread out as you go along and get bigger and bigger But notice that in the beginning here some of them are close together, so we've got actually a pair of consecutive Powers, so separated only by one, I guess here We have separated by two and three and four and so on so there's some separation here But what Catalan conjectured and this is a question people have been interested in for hundreds of years actually is this particular pair of eight and nine so the fact that Three squared minus two cubed is equal to one or really that you have any two powers which differ by one No other example of that was known and so what Catalan conjectured is that this is the only time that this happens that you have two powers whose difference is exactly one But it was unknown if like a gajillion and four Yeah, that's exactly right so we didn't know for a long time whether or not this conjecture was true so back a few decades ago We did know that if it wasn't true there would be only finitely many exceptions So this tends to be easier to show in general right that okay? I know that there's a really large number of things happening or only finitely many times it happens But to show that there's only one is what we didn't know until recently so it's known Yes, so this is what was proved by this mathematician Mihăilescu. In fact, These are the only consecutive perfect powers, okay, so this notion that okay? They probably kind of spread out as time goes on is it least true if spreading out means bigger than one I mean the proof of this is really advanced and so okay We don't have time for I don't know the next couple of years to go through it but I want to tell you at least about sort of a special case of this and Something that's been known for a while But it gives you an idea of kind of the type of manipulation you can do to understand this kind of problem. all right, So solving this equation showing that there's only one solution and that's it is pretty hard But let me show you a special case: let's look at the case of x squared minus Y cubed equals one so for example that has some solution We know because we have these consecutive powers, so we're really just asking about a square differing from a cube by one okay So why doesn't this have any other solutions? Well, here's what you want to do: general rule adding and understanding factors at the same time is hard That's why Fermat's Last Theorem is hard, that's why this kind of question is hard. So let's change it to a question about multiplication. All right, so how do we do that? Let's move things to the other side, so let's say instead We're gonna solve this equation. Now why have I just made my life better? The reason why is because this thing breaks up into two factors Any time you see a difference of squares like this, you can factorize it in this way, and so this just changes from a very hard problem about addition and multiplication interacting together to just multiplication. Okay, so now why is this better? Why do I find out whether or not this has solutions? Here's the idea: the factors of Y. Any number that divides y has gotta divide one of these two things and so let's ignore two for the moment Let's pretend that Y is odd, so two doesn't divide Y So if Y is odd that means then that any factor of y has to divide one of these two things But it can't divide both of them at once and the reason why is that whatever these two numbers are, they differ only by two So for the same reason they can't have a common factor other than possibly two. so if Y is odd all the factors of Y either divide X minus one or a factor of Y divides X plus one, but not both at the same time Which means that both of these numbers have to be cubes. So these are both cubes because any number which divides Y We know there's at least three copies of But all of those copies have to go to the same one of these two numbers, and so we've got two cubes Which are only two apart. Right so the difference between these two cubes is only two, but that never happens with cubes, right, because we know what the cubes are 1 8 27 and so on and that for sure spreads out as you go along and so you can't have any cubes which are separated by two And so you can't have any solution to this equation (at least if Y is odd). -What if Y is even? Well then it's a little bit harder, but not that much -You haven't done the whole proof No, so in general of course the proof is not via this plan although changing from addition to multiplication is the important part of the proof actually. An important part of the proof. But this is just sort of the smallest tiniest piece the step in the right direction towards proving this kind of thing. Hi everyone thanks for watching this video, I'm just working on Brilliant's problem of the week, which triangle has the greatest area. I can't decide if this is a trick question or do they have the same area let me get 4c ah It says it's okay. I got it wrong, but it says here And this is true getting stumped is part of learning and now we get to continue and discuss the solutions Yeah, see I probably should have done better working out like that person did, but you know we're recording here in a hurry now Brilliant dog is a fantastic problem-solving website It's also the sponsor of today's video in case you hadn't noticed and the thing I like about brilliant is it's full of all sorts of quizzes and courses and curated lessons that doesn't just Tell you stuff so that you'll know it it helps you understand that it helps you go deeper And I think that's a really important thing you can watch loads of videos and learn lots of new things but to really Understand that sometimes you actually have to do the problems have someone hold your hand as you go through it And that's what's great about brilliant. They obviously cover loads of mathematics They also cover science. They also cover computer science. There's physics. There's all sorts of things here You really should have a look by the way if you enjoyed today's video with Holly I'm told the open problems group here on brilliant could also be a great place for you to be hanging out So what are you going to have a look at that? I'll include a link down in the video description now while I get back into these triangles and figure things out What you should do is go to brilliant org slash numberphile you can sign up to brilliant for free? But if you use the slash numberphile, I just mentioned and there'll be a link in the description You'll also get 20% off a premium membership go and have a look and don't just get told stuff understand stuff There is a difference
I please tell me there's a Numberphile2 video with just a bit more detail. I know I can't handle the whole proof, but just a little bit more would be nice to chew on.