 I'm going to be talking about the integer 26253741264076800. So the first question is, why is this integer interesting? Well, one thing you can notice is that 640320 cubed. Well, that's not terribly interesting. There are lots of cubes. The real reason why it's interesting is if you work out e to the pi root 163. This turns out to be 262537412640768743.9999999999999999925 and so on. So we've got this spectacular sequence of nines just after the decimal point. In other words, this is amazingly close to being an integer. It's so close that an ordinary pocket calculator can't check this, and you need special software. There's an online calculator, which does very high precision called free 42 that I'll try and put a link to in the video description just in case you want to check this. So Martin Gardner in his mathematical games column once had a mathematical hoax, which I can show you. His columns were reprinted in a series of books, and this book is Time, Travel, and Other Mathematical Bewilderments. And on page 126, he says, the number theory, the really exciting discovery is when you raise e to the power of pi root 163, the result is an integer, which isn't true. And then he talks about Ramanujan, who was one of the most extraordinary mathematicians of all time. He was entirely self-taught and grew up in poverty, but somehow managed to become one of the world's greatest mathematicians. And Ramanujan did indeed have a note in the quarterly journal of pure and applied mathematics that, in fact, we'll look at later. And this is the correct value of e to the pi root 163 as far as it goes. But then Martin Gardner claims the nines continue, which isn't correct. So what's going on? Is this just a meaningless coincidence? I mean, if you get enough numbers, then some of them are going to end with lots of nines just by coincidence. Well, let's check a few other examples. e to the pi root 67 is equal to 147197952743.999998. How about another one? e to the pi root 43 is equal to 8, 84736743.9998. So something is obviously going on. There's something else a bit weird. If you look here, they all end in the same three digits, 743. So we seem to be getting the number 744. And this is very odd, because this only seems to be happening because we are working in a base 10 number system. So there shouldn't be any relation between these, even if they are all integers. So what happens if you do these in a different base? Well, really oddly, even if you write out these numbers in several other bases, the last three digits are quite often 743 in whatever base you're working in. So there's something odd going on about that. Remember the number 744. So what's the explanation for all this? Well, this is related to something called the elliptic modular function. And I'm going to give you the definition of the elliptic modular function. And I warn you in advance, the definition of the elliptic modular function kind of looks like a hoax. It looks like some random nonsense thought up by somebody who was either trying to play a joke or had no idea what was going on. So you take 1 plus 240 times the sum over m greater than 0. You sum over n greater than 0. And you take m cubed times q to the mn. Here, q is equal to e to the 2 pi i tau, just in case this wasn't complicated enough. And this is by no way near complicated enough. So let's cube that for no particularly good reason. And let's make it more complicated. We'll divide it by q. And then, I don't know, let's divide it by 1 minus cubed for 24 for no particular reason. And 1 minus q squared for 24 and 1 minus cubed for 24. And let's just keep going. As you see, this is just a completely ridiculous thing to do. So what on earth is the point of this function? Well, if you look at the function f of tau equals e to the 2 pi i tau, this is the simplest function, which satisfies f of tau is equal to f of tau plus 1. And see this by writing this by all its formulas cosine of 2 pi tau plus i sine of 2 pi tau. So these are the trigonometric functions cos and sine. And j of tau is equal to j of 1 plus tau. But it's also equal to j of minus 1 over tau. And e to the 2 pi i tau is the simplest function satisfying this property here. And j turns out to be the simplest function satisfying these properties here. Well, apart from constants, of course, which is a bit silly. It's always struck me as really extraordinary how this really simple condition on a function, the simplest function, turns out to be this extraordinarily complicated mess. So what does j of tau look like? Well, first of all, you can expand it as a power series of q. And as you might expect, it looks like a real mess. So it looks like this. And well, what do we see here? Well, first of all, we've got this number 744. And I told you to remember 744 from the previous sheet because it was appearing at the end of all these numbers. And here it is appearing again, which is certainly rather fishy. There's also this number 196884 that I want to remember because it's going to appear later as well. I can also show you some pictures of the elliptic modular function. These come from a rather old book by Yonka and Emda. So it's their book Tables of Functions with Formulae and Curves. And it was first published in 1909. And what you should remember when you see this picture is that 1909 was many, many years before computer graphics programs were out. And these rather spectacular pictures were drawn by hand by an expert draftsman. Heaven knows how many calculations they had to do to get them. So this is the graph of the elliptic modular function. Well, it's not quite. It's actually a graph of the absolute value of the elliptic modular function on the upper half plane. Well, that's not really right either. It's actually a graph of a different elliptic modular function, but the real elliptic modular function looks kind of like this. See, all these things sticking up are poles. They're called poles for obvious reason. If you've ever taken a complex analysis course and wondered why poles were called poles, this picture should make it pretty obvious. So here's a second view from the other side of the elliptic modular function. There should really be infinite numbers of poles getting smaller and smaller, but I guess they're draftsman ran out of patience after drawing five of them. So what does the elliptic modular function have to do with e to the pi root 163? Well, let's look at j of minus one plus root minus one, six, three over two. And this turns out to be 262537412687680000. This is exact. And this is really bizarre that it's exactly an integer because the definition of the function was a complete mess and you would normally expect the value of any such function to have a lot of rubbish after the decimal point, but this turns out to be exactly an integer. So Martin Gardner wasn't so far off in his hoax that there's a very similar funny looking number that is exactly an integer. Well, why is it an exact integer? Well, I'll explain that a little bit later. Well, I won't explain it a little bit later, but I will comment on it a little bit later. Now, if you expand j of tau as q to minus one plus 744 plus one, nine, six, eight, eight, four, q and so on, with q is equal to e to the two pi i times this number here, it turns out that what you get is minus e to the pi root 163 plus 744 minus one, nine, six, eight, eight, four, e to the minus pi root 163 and so on. So what have we got here? Well, first of all, we've got this number e to the pi root 163 that we started, that we mentioned in the first sheet. Then we've got this funny number, 744 that keeps turning up and then we've got this random garbage and this random garbage is absolutely tiny. Now, you may think it doesn't look tiny because one, nine, six, eight, eight, four looks really big. However, pi times 163 is a pretty large number and e to the minus a large number is absolutely tiny. I mean, it's sort of 10 to the minus 12 and these numbers here are even smaller. So what we see is that if we rearrange this, we find e to the pi root 163 is equal to 26253741268076800 coming from that plus 744 plus something tiny rather than minus something tiny. And this is why e to the pi root 163 is so close to an integer. It's because this term here is extremely small. Okay, well, what's the big deal about 163? So why 163? If you replace 163 by most integers, you don't get anything like an integer. You get something that looks much more complicated. But it turns out that if you look at j of a plus root minus b over c where a, b and c are integers, there is an exact expression for this. And sometimes this exact expression is an integer. Well, what happens when it isn't an integer? Well, that's what Ramanogen's paper is about. So you remember on the first piece of paper I'd mentioned Barton Godin as a hoax where he mentioned Ramanogen's paper. And here's a copy of Ramanogen's paper. If you want to look this up, it's in a copy of Ramanogen's collected papers. And if you go to here, let me just magnify it again so you can see it more clearly. You see Ramanogen is indeed discussing numbers of the form e to the pi times root something. He's got e to the pi root 58 is something, something, something 0.999. And in the rest of his paper, he works out cases of this where you don't get an integer. And here are some examples of it. So you see he's got several pages and I'll go on to some of the more spectacular ones. So here he's got several pages of examples. And let's just go down here where he's got a particularly spectacular example. So here he's got an example of a special value of one of these functions where it takes four lines to write out and involves large numbers of square roots and so on of various random integers. So clearly Ramanogen had a lot of spare time on his hands for working these out. Of course, he was doing this way before computers were around, he did all this by hand calculation. So I still haven't really explained why 163 works better than other integers. Well, this comes from something Euler did. So Euler had this rather famous polynomial x squared plus x. This was x squared minus x plus 41, which is prime for x equals one, two, up to 40. So for x equals one, it's 41. For x equals two, it's 43. And for x equals 40, it's 1601. So it's very good at producing primes. And there are some other polynomials like this x squared minus x plus 17 is prime for x equals one, up to 16, where that goes from the prime 17 up to 257. And x squared minus x plus 11 is prime for x equals one, up to 10, where it goes from 11 to 101, I guess. And it turns out that these polynomials, these are just quadratic polynomials and they have discriminants, b squared minus 4ac. And the discriminant here is minus 163. And here the discriminant is minus 67. Here it's minus 43. And now you remember these numbers 163, 67, and 43 were the numbers that appeared on my first sheet of paper in this exponent here. And it turns out that if this polynomial is often prime, then e to the pi times root minus the discriminant is nearly an integer. And j of, that follows because j of minus one plus root d over two is an exact integer. So you can ask, are there any numbers bigger than 163 for this happens? Well, this was a famous open problem that Gauss asked. And Hager and Stark in about 1950s and 1960s showed that in fact, there are no numbers bigger than 163 with this funny property. Okay, there's one other weird property of the elliptic modular function I want to mention. So you remember j of tau was equal to q to the minus one plus 744 plus one, nine, six, eight, eight, four, q and so on. And I asked you to keep an eye on this number, one, nine, six, eight, eight, four. Well, this number turns up because there is a mathematical object called the monster group discovered by Grice, which lives in one, nine, six, eight, eight, three dimensions. You can imagine there's a sort of huge snowflake living in one, nine, six, eight, eight, three dimensions. And the monster group is the collection of symmetries of this snowflake. And the number of elements it has is completely ridiculous. It has about eight times 10 to the 53 elements. So there's a video by three blue, one brown, who's already talked about this number. So I'm not going to repeat him and I'll try and add a link to his video, although this is probably pointless because my guess is most people seeing my video probably came by watching his video, but anyway. So the monster group is very mysterious related to the elliptic modular function. So John McKay noticed these two numbers were very nearly the same and that turns out not to be a coincidence. This observation is one of the few pieces of higher mathematics that has appeared in a comic book. So here we have a copy of the book Planetary, Volume One by Ellis and Cassaday. And they actually have a picture of the monster group here. So let me just zoom in on this. So, and you can tell this is actually the monster group by what they say about it. So down here, they mention that, you know, it's a theoretical snowflake in one, nine, six, eight, three, three dimensional space. And if you were paying careful attention, they've got a misprint here. It should be one, nine, six, eight, eight, three dimensional space. And then they say the total number of rotations are equal to the number of atoms making up the earth. Well, that's more or less correct. The number of elements of the monster group is indeed more or less the same as the number of atoms making up the number of making up the earth. And when I say more or less the same, I mean, wrong by a factor of a few thousand or so, but whatever. I guess strictly speaking, the picture of the monster is also completely unlike the monster, but so they've got the order wrong and they've got the dimension wrong and they've got the picture wrong. Yeah, well, apparently comic books are not a reliable source of information about mathematics. So I will just finish by giving some further things you can find out more about. You may have noticed, I did not actually give an explanation of anything in this talk. If you want to see an explanation of what's going on, it requires a certain amount of graduate level mathematics. So first of all, if you want to know what the elliptic modular function is, there's this very nice book by a post or called Modular Functions and Dirichlet series in number theory and he explains the elliptic modular function. Here is his definition of it and describes its basic properties. If you would like to know why the elliptic modular function is sometimes an integer, the Bible of this subject is the book introduction to the arithmetic theory of automorphic functions by Shimura. So here, for example, he has a theorem saying that under certain conditions, J and Z, that's the elliptic modular function, is an algebraic integer. This book is even harder than a postals book. And finally, if you would like to know more about the monster simple group, there is a construction of it in this book by Conway and Sloan called Sphere Packings, Lattices and Groups. So if you look at chapter 29 by Conway, he has a construction of the monster in 1, 9, 6, 8, 8, 4-dimensional space. Okay, so that's all about the number, whatever.