163 and Ramanujan Constant - Numberphile

ALEX CLARK: And what's interesting about this problem is that it was actually solved by what might be called an amateur mathematician, someone that wasn't officially a mathematician. So I first encountered 163 when studying a little bit of basic number theory. And it came up in the most mysterious way that fascinated me because it's the last number in a short list of numbers. Even though it seems very innocuous, it has some very mysterious properties. So you may have heard about factoring numbers. You may not remember exactly what that means. So let's just review that. If you start out with a whole number, say 12, you can write it as a product of numbers in many different ways. So you could think of 12 as being 6 times 2, or you could think of it as being 4 times 3. But if you want to write it as a product of prime numbers, there's only one way to do it, up to the ordering in which you write the numbers. So 2 and 3 are both prime numbers. So every whole number can be written in exactly one way as a product of prime numbers, up to the order in which the numbers are written. BRADY HARAN: Any number? ALEX CLARK: Any number. BRADY HARAN: All right. I'm going to give one, you ready? ALEX CLARK: No. OK. BRADY HARAN: 20. ALEX CLARK: That is going to be 2 times 2 times 5. And each one of those is a prime number. A mathematician named Gauss was trying to identify all of the perfect Pythagorean triples. So what are these? These are the whole numbers a, b, and c that satisfy this equation. And what this means geometrically, the reason these would be called Pythagorean triples, is that these numbers are the side lengths for a right triangle. The most common example of a perfect Pythagorean triple would be 3, 4, and 5. And so the natural question is, well, we can find specific examples. How do we find an exhaustive list of all the examples of Pythagorean triples? And what Gauss noticed is that well, if we take the idea of factorization of ordinary whole numbers to a new number system, that we can make some progress. And so the way that he created this new number system was to take all of the whole numbers and add i, sometimes denoted this way. So he takes all of the whole numbers, that's what this zed corresponds to, and he adds i. So this consists of all of the numbers that can be written this way, where the a and the b are whole numbers. And i is the square root of minus 1. Well, what he noticed is that he could factorize this expression, if he just had i in a way that can't be done in the ordinary number system. So if we take a squared plus b squared, you might remember from school that this can't be factorized in any nice way using ordinary numbers. If it were, instead, a difference of squares, there would be a nice way to do it. But if we allow ourselves to use i, we can actually write this. And what's more, the special property that each number can be factorized into primes in exactly one way carries over to this new number system. Now you have to expand your notion of what a prime is because you've got now new numbers. So they may or may not be primes. And it could be that what used to be primes are no longer primes in the new system. But what he also discovered was that if we adjoin other numbers instead, we might not get the same result. So for example, in this number system, this is just all the normal numbers, except that we now allow ourselves this one new number. So this one still is a number system, but it has the unfortunate property that it doesn't have unique factorization. So just to give you an example. In this number system, we have the number six, which could be written as a product 2 times 3. But, at the same time, it can be written as this product. This is a problem because here we have one way to decompose it into a product of primes, and here we have another way that's completely different, using completely different primes. It's not just a matter of rearranging the ordering. BRADY HARAN: So it worked for the square root of minus 1. It didn't work for the square root of minus 5. Where are we going here? ALEX CLARK: So what he observed was that it doesn't work for all numbers, but he did identify quite a few for which it did work. And specifically, he identified a list of numbers. We still do have the unique factorization, provided we choose one of the following numbers, d. Well, one-- these are the numbers for which we get unique factorization. And then there's a big jump to this number. BRADY HARAN: So that one works. ALEX CLARK: And this one works. As you can tell, it's quite a bit removed from the previous one. And what Gauss conjectured was that this was the last one. That for some reason, with this number, the list stops. And no matter what other number after this you add, you no longer have unique factorization. He conjectured that this was really the way it was, that it was true, but he could never actually prove it. BRADY HARAN: I can see why that number intrigued you now. ALEX CLARK: The problem is to try to actually show that this is the last number for which this happens, that this list is exhaustive. That there are no other numbers of this sort that one can add in order to get unique factorization in the new number system. BRADY HARAN: Tough nut to crack then, was it? ALEX CLARK: It was a very tough nut to crack. And what's interesting about this problem is that it was actually solved by what might be called an amateur mathematician, someone that wasn't officially a mathematician. So a German by the name of Heegner actually found a proof in the '50s. And he did actually publish his result, but it wasn't accepted by the mathematical community for several years. And then in the '60s, some well-established mathematicians, a British mathematician by the name of Baker and an American by the name of Stark, found other proofs which were generally accepted. And then, after a couple of years, one of those mathematicians, Stark, went back to what Heegner had done, analyzed it in more detail, and realized, hold on a second, he did actually have a proof. It was actually OK with just some minor issues. BRADY HARAN: So, and what did Heegner find? Was Gauss right? ALEX CLARK: He was right. This is the end of the list. And mysteriously, these are the only ones. And there are some other interesting and mysterious consequences of this that you wouldn't guess just from the way these are presented. So one of these is the fact that there are certain numbers that Ramanujan had identified as being very close to whole numbers. So among these are this number, this number. And who knows how he managed to determine this. But Ramanujan did observe that these were close to whole numbers, surprisingly close. And as a sort of joke in the '70s, the recreational mathematician, Martin Gardner, wrote in one of his columns that Ramanujan's conjecture had finally been proven, namely that this was a whole number. So here we've got the number e. Here we've got the number pi. And this is taking e to this power. So e is anything but a whole number. It is as far from a whole number as you could expect to be, likewise for pi. The square root of 163 is also not a whole number. So it would be extremely surprising if you took this combination and somehow wound up with a whole number. And you don't. But he was able to fool people because it's so close to a whole number that, with the techniques that most people would have had available in those days, they wouldn't have been able to tell the difference. It's extremely close to a massive whole number. Makes it very difficult to check exactly what it is. But it turns out to be different from it by just ever so slightly much. BRADY HARAN: Is the fact that e to the power of root 163 pi is very close to a whole number in any way related to all this stuff we've already been talking? ALEX CLARK: Yes. In fact, the fact that you get all of these from those is directly related to that fact, not in an obvious way. It's quite involved to see exactly how that works, but it is, in fact, related to that. And I'm not even sure how it is that Ramanujan was able to identify these. I'm not sure that he knew about these properties, but he did identify those as being very close to whole numbers. Very strange. It's not something that I think ordinary people would ever have expected. And who knows why it was that Gauss conjectured that this would be true? Because one might also expect that, since you have this big gap between 67 and 163, that the next one might just be very far along. And before people had computers, how would they have known? How could they possibly check? So I wouldn't have guessed that. I don't know who would, other than someone like Gauss. But it's a very mysterious fact, but one that has now been established. BRADY HARAN: That's why you think 163 is a pretty cool number? ALEX CLARK: It is a very cool number. BRADY HARAN: Is it like your PIN number or the combination on your briefcase? ALEX CLARK: No, no, but it is certainly one of my favorite numbers.