[Tony]: I thought we’d have a look at some of the shortest papers that have ever been written. One of the sort of most famous ones was this one here by Lander and Parkin. Which you can literally see, look how short it is! There’s virtually nothing there. But the content’s really good because it actually disproves a very long-standing conjecture that had held for hundreds of years. [Brady]: So now we’re just disproving with an actual example? [Tony]: Exactly, they were just proving the Euler conjecture, and what is the Euler conjecture? Well it’s kind of related to Fermat’s Last Theorem. Fermat’s theorem, what does it say? You’ve got two numbers, a1 to the k plus a2 to the k equals b to the k. And basically it says there are no integers solutions for k greater than two. Okay, that’s Fermat’s Last Theorem. Euler’s conjecture was a generalization of this. Basically, you just take n of these guys, and again, now you’re saying, Euler conjecture would be there are no integer solutions for k greater than n. No integer solutions for k greater than n, so of course, Fermat’s Last Theorem is the case of n equals two. This is the general case, right? But this is not true, right? This paper proves it. Basically, in this example, n is four, but k is five. Disproven, there you go. So, in a nutshell, it disproves it. But this doesn’t have the record anymore. I don’t know if it ever did have the record. It’s sort of a classic short paper that achieves quite a lot. The record is, I think, currently held by this paper, by your friend, John Conway, and somebody called Alex Sofer, I think it is, I think that’s how you pronounce it. Okay, so they ask a question in the title, so I think that’s a bit of a cheat. And then, so they’re basically asking about how many equilateral triangles you need to cover an equilateral triangle of a given size. Basically, the paper just says n squared plus two can, and then it has a picture of their solution. So it’s kind of got two words in it, right? They had the following question, right? So you take an equilateral triangle and you imagine it’s got a length of side n, and then you could ask, well, how many triangles of side one, you know, so unit side triangles, equilateral triangles, do I need to to fill it, to cover it? You know the answer. It turns out to be n squared, it’s quite easy to see that. Show you by example. If you’ve got side two, so we’ve got side two, we can fill it with four equilateral triangles, like that. So, these have all got side one and four of them fill the equilateral triangle you started with. [BRADY]: Which is two squared.
[TONY]: Which is two squared exactly, right? And it generalizes and the answer is n squared for one of side n. Okay, so what these guys want to know is what if we make it a little bit bigger than n? Not n plus one but just a little bit bigger than n,
so n plus epsilon. How many do you need then? Okay, so you might think, “I know I’m going to need more than n squared.” Because n squared would cover the original one, right? So I’m gonna need more than, so you’re going to need at least n squared plus one. So they were asking, well can you do it with n squared plus one? And they didn’t actually answer the question. They say actually you need n squared plus two. How do they show that? Well, what you do is you sort of stop it at n minus one. So you draw, consider n minus one there. Okay and then you’ve got a little bit there of one plus epsilon. So you’ve got a little of one plus epsilon bit there. Now this thing, this big thing in here, is going to need n minus one squared to fill it. ok ? Now, what they do is they say right, what I need to fill this little bit left over here? Well, the way they do it is you know this has got length
n plus epsilon that way. Okay, so you squeeze in. you’re not going to cover these edges with n edges of the triangle, right? you’re going to need at least n plus one. Okay, so you squeeze in n plus one of them. Like so. [BRADY]: Maybe overlapping. [TONY]: They definitely have to overlap, right? Because this is n plus epsilon, so it’s n plus a tiny bit. And you’re going to squeeze it n plus one of them, so they’re definitely gonna overlap, So you get n plus one of them there. Okay and you fill in the top bits. To fill in the top bits so on and so forth, you need another n, and then you can do it. So how much have you got total now? In total you’ve got n minus one squared plus n plus n plus one. Okay you work out what that is, this is… n squared minus 2n plus one plus 2n plus one, which is n squared plus two. [BRADY]: The extra bit you need could be less than two, they’re just showing it’s– [TONY]: Yes, so that’s just what I think about it. I don’t think they even answered the question. The question is can n squared plus one unit equilateral triangles cover an equilateral triangle of side n plus epsilon? They don’t say yes or no, they just say n squaredplus 2 can. Maybe they could have just said “maybe” (laughs). I don’t know, they don’t answer the question! If you look in other fields there are shorter ones there. I mean, I found some others. There’s this one. It’s not maths, but in the applied behavior analysis, it’s “the unsuccessful self-treatment of a case of writer’s block”. Of course, there’s nothing there. There’s another one: comprehensive overview of chemical-free consumer products. This is in a chemistry journal, and there aren’t any. Actually, this didn’t just didn’t appear in the journal. They didn’t publish it, but they did include it for amusement purposes. There’s plenty of short papers out there. There’s also short abstracts. That’s another thing you can you can look at. This is one that caught my attention. This is a physics one. Can the apparent superluminal neutrino speeds be explained as a quantum weak measurement? There’s a classic thing in, you know, within academia, that if your title has a question mark at the end of it, then the answer is probably not. And indeed their abstract is “Probably not.” I would actually say definitely not, because we now know that there are no faster-than-light neutrinos, that that was just a screwed-up experiment. There’s an even shorter abstract there. There’s another one with a question. Is the sequence of earthquakes in southern California with aftershocks removed Poissonian? So this is a mixture of maths and, I guess, geography. But the answer isn’t “probably not”, it’s “Yes”! So that sort of goes against the usual rules of the game. But this is all just a bit of fun, right? But I think what’s more, what would be better is like, it’s not so much how short the paper is but, in a way, impact per word. So short paper that achieved a lot. Okay now I, this is clearly open for debate, but i think that the candidate for this is this bad boy by the beautiful mind, John Nash. So this is a very, very short paper, which it starts there and it finishes there. Okay, so it’s a page long and this basically started off game theory. This is really one of the main papers
underlying economics. He basically ultimately won the Nobel Prize for economics for this paper and other ideas related to it. So I think in terms of impact per word this is probably the winner. John Nash’s thesis is famously short as well. I’ve got a copy of his thesis here. Non-cooperative, of course, it’s about game theory,
Non-cooperative Games. It’s only like 26 pages long.
It’s like wow, you know what I mean? This thesis won a Nobel Prize and it’s 26 pages long. There’s a bit of an urban myth. If you do a maths degree, as I did, there’s an urban myth that goes around the students is to, there’s this part, this PhD thesis that someone did once and it’s like a page long and it proved some result from,
that hadn’t been known for hundreds of years, and it just did it in one page and there you go, bang,
PhD thesis came out. I think that is a myth but I would love it if somebody found it out that it wasn’t. I’ve had a little digging around and I don’t think it is. I think in terms of short theses, big impact, you’d do well to be good ol’ John Nash. [Brady]: This episode has been supported by audible.com, and if you’ve got some reading to catch up on, an audiobook is a great way to do it. Among Audible’s vast collection of titles is the acclaimed book A Beautiful Mind, the biography of John Nash,
who we’ve just been discussing. It’s written by Sylvia Nasar. Or if biographies aren’t your thing,
but you still want something with a bit of a math twist, I also highly recommend The Humans, a novel by Matt Haig, and one of my favorite stories of the last few years. But whatever the sort of thing you like, I can guarantee Audible is going to have it. And you can also get started with a free 30-day trial by going to audible.com/numberphile. They’ve got a great store, a cool app, really excellent features like Whispersync for voice so you can synchronize your audiobook on Kindle without losing your place. And if you do give that free 30-day trial a go, make sure you use the address audible.com/numberphile just so they know you came from the channel here.