1 00:00:22,940 --> 00:00:27,040 Thank you very much. I thought we could begin with some counting. 2 00:00:31,970 --> 00:00:36,830 Of course, maths isn't only about counting, but I'm interested in number theory properties of whole numbers. 3 00:00:37,670 --> 00:00:44,990 And I particularly like this way of visualising the numbers as it starts to give us an insight into Prime Factorisation. 4 00:00:45,020 --> 00:00:50,510 And I want to talk all about prime numbers today, so maybe you can start to notice some patterns here. 5 00:00:50,660 --> 00:00:57,950 And I want to talk about patterns. This isn't such a convenient way of of looking for some of the patterns that I want to do, though. 6 00:00:58,370 --> 00:01:02,209 So I want to arrange the numbers on a slide like this. 7 00:01:02,210 --> 00:01:05,720 So the idea here is we can start seeing patterns. 8 00:01:05,990 --> 00:01:12,620 And as mathematicians, our job is to look for patterns, to look for structures, and then to understand what's going on behind the scenes. 9 00:01:13,010 --> 00:01:17,300 So there's lots of patterns here that we might notice, and I'm not going to be able to talk about all of them today. 10 00:01:17,960 --> 00:01:21,170 So an example might be if you look carefully. 11 00:01:22,220 --> 00:01:24,920 So these columns, the four column, the six column, 12 00:01:24,920 --> 00:01:31,430 the eight column and the ten column don't seem to have any prime numbers in from the basis of what we can see on the slide. 13 00:01:32,350 --> 00:01:35,889 But I'm not going to be satisfied that I'm interested in what would happen if we kept going. 14 00:01:35,890 --> 00:01:38,590 If we continued this slide further, what would happen? 15 00:01:39,010 --> 00:01:45,220 And if you look carefully, these four, six, eight and ten columns, all the numbers in those columns are even, I think is a prime number. 16 00:01:45,400 --> 00:01:52,600 It's only is a one in itself. So if you've got an even number bigger than two, it's divisible by one by itself, but also by two. 17 00:01:52,720 --> 00:01:56,379 So it can't possibly be prime. So if two is the only even prime. 18 00:01:56,380 --> 00:02:01,180 So in its column there, we're not going to find any more primes in that column, no matter how far we go. 19 00:02:01,450 --> 00:02:04,120 And these four, six, eight and ten columns really are empty. 20 00:02:04,360 --> 00:02:11,040 So that's not the most complicated example, but it gives you a flavour of noticing a pattern and then trying to understand what's really going on. 21 00:02:11,050 --> 00:02:16,030 Why are we seeing that? And the kinds of patterns I want to focus on today. 22 00:02:16,030 --> 00:02:18,490 It's to do with gaps between prime numbers. 23 00:02:18,820 --> 00:02:26,320 So, in fact, that observation about even primes shows us that if we look at two and three, two or three consecutive numbers and that both prime, 24 00:02:26,680 --> 00:02:32,110 but that the only example of two consecutive numbers that are both prime because if we pick any two consecutive numbers, 25 00:02:32,110 --> 00:02:38,260 one of them is going to be even and we just said two is the only even prime number one, by the way, is not prime, 26 00:02:38,590 --> 00:02:43,020 because it turns out that that leads to a better definition at least of better mathematics that way. 27 00:02:43,030 --> 00:02:44,620 So we define one not to be prime. 28 00:02:45,130 --> 00:02:51,430 So if the only pair of consecutive numbers that are prime primes two and three and up near the top, these primes of bunched up quite close together. 29 00:02:51,430 --> 00:02:59,649 If we look further down, there's some quite big gap. So 83 to 80 9 to 1997, these primes are getting further apart and that makes intuitive sense. 30 00:02:59,650 --> 00:03:01,690 If I give you a really big number, a number this big, 31 00:03:02,140 --> 00:03:07,209 it's very hard for it to be prime because it's going to be divisible potentially by lots of smaller things. 32 00:03:07,210 --> 00:03:10,780 There are lots of smaller numbers that might divide it that would stop it being prime. 33 00:03:10,780 --> 00:03:15,070 It's very easy for seven to be prime because there aren't many smaller numbers that would stop it. 34 00:03:15,550 --> 00:03:20,410 But this big, there are lots of potential factors, so it sort of makes sense that these primes are getting more spread out. 35 00:03:21,810 --> 00:03:26,280 One question we might ask is whether we keep finding more and more primes. 36 00:03:26,430 --> 00:03:29,489 If we continue down, if I could fit more and more on this slide, 37 00:03:29,490 --> 00:03:33,390 would I keep finding new primes or at some point where we hit the biggest prime number? 38 00:03:34,510 --> 00:03:39,930 Those are two very different scenarios. Right. And one world. There's a biggest prime number from that point on. 39 00:03:39,930 --> 00:03:46,230 Nothing is prime. In the other scenario, no matter how far we go, we keep finding new prime numbers. 40 00:03:46,680 --> 00:03:50,190 And it's sort of intriguing which of those situations it might be. 41 00:03:51,480 --> 00:03:53,940 So I guess that intuition about if you're a very large number, 42 00:03:53,940 --> 00:04:00,629 it's very hard for you to be prime is not quite enough to guess which way it goes because maybe they're getting further and further apart, 43 00:04:00,630 --> 00:04:05,100 but they still keep happening. Or maybe they're getting so far apart and eventually they kind of stop. 44 00:04:05,100 --> 00:04:08,610 But it's sort of I can't quite make that intuition help me figure out which it should be. 45 00:04:09,120 --> 00:04:13,280 I could use a computer to help, but I see that's not going to get me very far. 46 00:04:13,290 --> 00:04:17,699 I can ask my computer, please tell me the largest problem you can think of and it goes work and clunks 47 00:04:17,700 --> 00:04:20,670 for a fortnight and comes back and says he got his very large prime number. 48 00:04:21,420 --> 00:04:27,690 And that tells me nothing at all because maybe it's found the largest prime number in the world. 49 00:04:28,200 --> 00:04:30,749 Or maybe if it thought for another month it would think of another prime number. 50 00:04:30,750 --> 00:04:34,830 So computer is not going to help there, but happily, mathematics will help. 51 00:04:35,250 --> 00:04:41,430 And this goes right back to the ancient Greek. So this is a theorem, this is a mathematical statement that we can prove to be true. 52 00:04:41,700 --> 00:04:45,660 And the theorem is that there are infinitely many primes. There is no biggest prime number. 53 00:04:46,350 --> 00:04:50,320 And for me, the certainty here is what's really important. 54 00:04:50,340 --> 00:04:56,999 So when the radio comes on in the morning with the headlines, I don't have half a percent of my brain thinking, Oh, 55 00:04:57,000 --> 00:05:03,630 I wonder whether today they'll find the largest prime number because I am absolutely, totally certain there is no largest prime number. 56 00:05:03,810 --> 00:05:10,320 And the reason that I'm totally certain is because we have a proof. So for me, proof is really important part of what we do as mathematicians. 57 00:05:10,920 --> 00:05:14,400 And I want to share this proof with you because proof is really important 58 00:05:14,490 --> 00:05:17,790 because this is an especially beautiful proof and it's a very well known proof. 59 00:05:17,790 --> 00:05:22,830 So lots of you will be familiar with this proof already, but I hope that like me, you will enjoy thinking about this again. 60 00:05:23,130 --> 00:05:28,680 And if you haven't seen this before, it takes a little bit of getting your head around first time, but it's worth thinking about. 61 00:05:28,680 --> 00:05:32,159 So this goes back to Euclid's sequence of books that he wrote. 62 00:05:32,160 --> 00:05:39,569 The Elements is kind of textbooks that he wrote more than 2000 years ago, and he had this beautiful idea for proving this. 63 00:05:39,570 --> 00:05:44,820 So he said, Let's do a thought experiment. So secretly we think there are infinitely many primes. 64 00:05:44,820 --> 00:05:50,250 There is no biggest prime number. Let's imagine we're in a parallel universe where there is a biggest problem. 65 00:05:50,250 --> 00:05:54,209 There are only finite, any many primes, and let's explore the consequences of that. 66 00:05:54,210 --> 00:06:00,350 And it turns out you reach an impossibility. So if we're in this parallel universe where there are only finite, 67 00:06:00,360 --> 00:06:05,820 how many points I can take very large pieces of paper, and on them I can write all the problems in the world. 68 00:06:06,060 --> 00:06:11,130 So my list starts two, three, five, seven, 11, whatever. And I keep going, keep going till I get to the largest problem in the world. 69 00:06:12,360 --> 00:06:15,930 And Euclid's beautiful idea was to take all the primes in the world. 70 00:06:16,920 --> 00:06:20,370 Multiply them together. I'd add one. 71 00:06:20,400 --> 00:06:23,700 So this is some enormous number, right? It doesn't matter what it is. 72 00:06:24,300 --> 00:06:29,730 What matters is how we've built it from these primes. So this enormous number must have a prime factor. 73 00:06:30,030 --> 00:06:33,209 Either it's prime itself or it's divisible by smaller prime every number. 74 00:06:33,210 --> 00:06:38,630 It has a prime factor. Once you go beyond one. So conveniently, I've got this list on this piece of paper. 75 00:06:38,670 --> 00:06:43,020 Of all the primes in the world, and I can just check which of them are factors of this large number. 76 00:06:43,440 --> 00:06:50,930 So just to divide this number. Uh, no, because it's two times some stuff plus one. 77 00:06:51,530 --> 00:06:56,799 So this. This we made one where we divide by two. Does three the next time on my list. 78 00:06:56,800 --> 00:07:01,120 Buy this number? No, because it's three times and stuff plus one. 79 00:07:01,120 --> 00:07:05,380 At least we made one. When we divide by three, does five divide by the same story. 80 00:07:05,410 --> 00:07:12,070 What it's done is he's built this number so that it's not divisible by any of the primes on the list. 81 00:07:13,130 --> 00:07:17,870 And that was supposed to feel slightly ill because somehow we've got this number that must have 82 00:07:17,870 --> 00:07:22,640 a prime factor and yet isn't divisible by any of the primes in the world that were on our list. 83 00:07:23,420 --> 00:07:26,479 So we call that a contradiction in that is this kind of impossibility, 84 00:07:26,480 --> 00:07:31,040 this absurdity that arose from supposing that there were only finite, any many primes. 85 00:07:31,040 --> 00:07:33,679 And that tells us that there must be infinitely many primes. 86 00:07:33,680 --> 00:07:37,970 So this little square box symbol is a symbol that mathematicians often use to the end of a proof. 87 00:07:39,020 --> 00:07:42,070 So, so more if you didn't take in all the details of that. 88 00:07:42,080 --> 00:07:46,150 But I think it's extraordinary when you think about it, we're proving something really difficult, right? 89 00:07:46,160 --> 00:07:53,900 We're proving that there are infinitely many prime numbers. And this argument is so elegant, so beautiful, that I can fit it on a whole slide. 90 00:07:53,910 --> 00:07:58,100 So I'm very glad they're infinitely many prime numbers. 91 00:07:58,100 --> 00:08:02,360 The world is a much more interesting place with infinitely many primes. There is no largest prime number. 92 00:08:02,630 --> 00:08:05,900 So if we go back, we can then kind of keep thinking about patterns here. 93 00:08:07,080 --> 00:08:11,610 I feel like I need to confess these. I've sort of slightly misled you. 94 00:08:12,630 --> 00:08:15,690 You know what I was saying about the problems getting further apart towards the bottom? 95 00:08:15,690 --> 00:08:18,690 Because if you're a big number, it's really hard to be prime. So the gaps get bigger. 96 00:08:20,240 --> 00:08:25,850 Well, here's the thing about the primes. Every time you think you understand what's going on, it's a little bit more subtle than that, 97 00:08:26,210 --> 00:08:32,000 because if I put another row on, I have to confess that 100, 100, 307 hundred and they're all prime. 98 00:08:32,780 --> 00:08:36,950 So, yes, they were getting farther apart. But then all of a sudden, here's a cluster bunched up all together. 99 00:08:36,980 --> 00:08:43,500 So what's going on? It is true that on average the primes are getting more spread out. 100 00:08:44,280 --> 00:08:48,299 That's a consequence of a theorem that was proved right at the end of the 19th century called the Prime Number Theorem, 101 00:08:48,300 --> 00:08:54,000 which gives us an insight into the distribution of the Prime. So on average, the primes are getting more spread out, 102 00:08:54,690 --> 00:08:58,950 but on average is really important because then you get these little pockets of problems 103 00:08:58,950 --> 00:09:03,330 that are very close and at least primes that are very close that I want to focus on today. 104 00:09:03,840 --> 00:09:13,710 So Primes, there's these problems that differ by two, like three and five, like 29 or 30, 107, 109 pairs of primes that differ by two. 105 00:09:14,160 --> 00:09:19,890 So we know there are infinitely many primes. So the question is, are there many pairs of primes that differ by two? 106 00:09:21,300 --> 00:09:28,500 And this is called the twin primes conjecture. So the conjecture, the prediction is that there are infinitely many pairs of primes that differ by two, 107 00:09:29,070 --> 00:09:32,940 and there are some sort of plausible reasons for thinking that this should be true. 108 00:09:32,970 --> 00:09:40,770 So one plausible but not super good reason is that I can ask my computer to find examples of large pairs of twin primes. 109 00:09:40,770 --> 00:09:45,260 Prime if it by two. My computer would go and think about it and they will come up with very large examples. 110 00:09:45,270 --> 00:09:48,660 We know that there are very large pairs of primes that differ by two. 111 00:09:50,080 --> 00:09:54,670 That's sort of not such a great reason, though, because, as I said, with there being infinitely many problems, 112 00:09:54,670 --> 00:09:57,190 there's always this risk that we find the largest part of two in primes. 113 00:09:57,430 --> 00:10:02,410 So a better reason is that we can kind of model the behaviour of prime numbers. 114 00:10:02,710 --> 00:10:06,370 So if I pick a number like 61, either 61 is prime or it's not prime, 115 00:10:06,370 --> 00:10:11,340 we we don't get to kind of toss a coin, but we can model the prime numbers by tossing a coin. 116 00:10:11,350 --> 00:10:16,989 So the idea is you have a biased coin and the probability of getting a prime 117 00:10:16,990 --> 00:10:20,590 is skewed a bit because the primes on average get more spread out later on. 118 00:10:20,600 --> 00:10:22,930 We know that it should be harder for a large number to be prime. 119 00:10:23,350 --> 00:10:30,009 So if you do this modelling very carefully, you can make a prediction for the number of pairs of twin primes, 120 00:10:30,010 --> 00:10:33,310 up to a million or billion or whatever your favourite number is. 121 00:10:33,790 --> 00:10:37,750 So we can make this kind of predicted estimate for the number of pairs of twin primes, 122 00:10:38,350 --> 00:10:42,280 and then we can check that against the data that our computer has given. 123 00:10:42,280 --> 00:10:47,410 That's where it's helpful to know that the computer has calculated these these large examples of twin primes. 124 00:10:47,860 --> 00:10:52,900 And the prediction matches the data we have computationally beautifully. 125 00:10:54,520 --> 00:11:01,110 But it's not a proof of anything because it's only this model. So we can't prove that this is exactly how the primes work. 126 00:11:01,120 --> 00:11:05,620 We don't yet know how to do that, but it's sort of very compelling evidence because in particular, 127 00:11:05,620 --> 00:11:12,009 it predicts that there should be infinitely many pairs of twin primes. So I thought it might be fun to have a go at trying to prove the twin primes 128 00:11:12,010 --> 00:11:21,980 conjecture because we should be ambitious if we take the idea of Euclid's proof. 129 00:11:22,000 --> 00:11:25,870 So Euclid said, Take all these primes of the piece of paper. Multiply them together. 130 00:11:25,870 --> 00:11:26,470 Add one. 131 00:11:28,040 --> 00:11:34,280 Nothing at all would change in Euclid's proof if instead we took all the points of a piece of paper, multiplied them together, and subtracted one. 132 00:11:34,820 --> 00:11:39,410 So we could do this. What if I take two, three and five plus few primes multiplied together, subtract one, add one. 133 00:11:39,830 --> 00:11:46,080 I get 29 and 31. That's a pair of twin primes. That's kind of nice that I could keep doing that, right? 134 00:11:46,080 --> 00:11:49,559 So I could take two times, three times five times seven and add one and subtract one. 135 00:11:49,560 --> 00:11:56,640 I get 209 211. The bad news is the 209 isn't prime. 136 00:11:57,850 --> 00:12:02,410 So Euclid's argument doesn't say the number we come up with must be prime. 137 00:12:02,590 --> 00:12:06,550 It says it's not divisible by any of the primes on the list we had. 138 00:12:06,730 --> 00:12:10,700 So as has happened here with 209, it is divisible by smaller primes. 139 00:12:10,720 --> 00:12:13,240 They were just too big to have been on the list. So. 140 00:12:14,490 --> 00:12:19,380 So unfortunately, we can't prove the twin primes conjecture by adding and subtracting one from this argument, 141 00:12:19,620 --> 00:12:21,380 which is a shame because that would be nice. 142 00:12:22,380 --> 00:12:29,910 But that's how much guys, sometimes mathematicians spend a lot of time trying to prove theorems without necessarily succeeding. 143 00:12:29,940 --> 00:12:39,030 So one of my mathematical heroes, a mathematician called Julia Robinson, who was a mathematician in the US in the 20th century and an administrator, 144 00:12:39,030 --> 00:12:42,510 asked her one occasion to describe her typical week for some form or other. 145 00:12:42,540 --> 00:12:46,410 So here's how Julia Robinson described her typical week Monday. 146 00:12:47,130 --> 00:12:50,130 Try to prove theorem Tuesday. 147 00:12:50,910 --> 00:12:55,680 Try to prove Theorem. Wed tried to prove theorem. 148 00:12:56,810 --> 00:13:02,800 Thursday try to prove their. Friday Third Falls. 149 00:13:03,760 --> 00:13:08,649 So we're sort of sounds very depressing, but it's not quite all like that. 150 00:13:08,650 --> 00:13:12,810 But mathematicians do spend a lot of time trying out some ideas that don't necessarily get them. 151 00:13:13,000 --> 00:13:17,950 So I like this because it gives us more insight into Euclid's argument, but it doesn't prove the twin primes conjecture. 152 00:13:18,190 --> 00:13:23,350 In fact, the reason the twin primes conjecture is still called the Twin Primes conjecture is that nobody has proved it. 153 00:13:24,340 --> 00:13:29,620 So we think it should be true. That's why it's a conjecture, but we don't yet have a proof. 154 00:13:31,180 --> 00:13:35,020 I spend a fair bit of time going into schools and talking to young people about maths, 155 00:13:35,020 --> 00:13:41,409 and I sometimes worry that I've given them the impression that maths is divided into things that we know how to do because 156 00:13:41,410 --> 00:13:47,470 mathematicians did them a long time ago and those mathematicians have long since dead and things that we don't know how to do. 157 00:13:48,580 --> 00:13:52,510 And that's not the case as all mathematicians are making progress on problems all the time. 158 00:13:52,690 --> 00:13:57,670 So what I want to tell you about today is some of the recent progress on this particular problem, the twin primes conjecture, 159 00:13:58,600 --> 00:14:04,270 because it gives us a bit of an insight into how progress is made and at least in some parts of mathematics. 160 00:14:04,990 --> 00:14:12,990 And the excitement happened back in May 2013 when a mathematician, Cody Tanjung, announced this extraordinary breakthrough. 161 00:14:12,990 --> 00:14:16,630 It got mathematicians super excited kind of progress on the twin primes conjecture. 162 00:14:17,200 --> 00:14:22,870 Zang managed to show that there are infinitely many pairs of primes that differ by at most 70 million. 163 00:14:24,170 --> 00:14:31,399 Now when you are aiming for the twin primes conjecture and you what pairs of points that differ by 270 million feels like a miss, right? 164 00:14:31,400 --> 00:14:38,350 I mean, it's a long way off. I agree. But it was the first time anybody had managed to prove any theorem of that type before. 165 00:14:38,360 --> 00:14:42,019 So 17 million is a lot bigger than two, but it's much better than infinity. 166 00:14:42,020 --> 00:14:48,350 It was a finite number that was the important thing. So there are infinitely many pairs of primes where the gap between them is less. 167 00:14:48,380 --> 00:14:54,980 They're equal to 70 million. Zhang was not a famous expert in this area of maths. 168 00:14:55,000 --> 00:15:00,340 He's had hardly any publications since his PhD. He found it very difficult to get a job in a university. 169 00:15:00,370 --> 00:15:04,089 He did, by this point, have a job as a lecturer at the university in the US. 170 00:15:04,090 --> 00:15:09,560 But he'd spent some time working for the Subway restaurant chain, doing various bits of books, but was doing all of that. 171 00:15:09,580 --> 00:15:15,850 He kept up with the research literature, so he really immersed himself in the area and that was what enabled him to do this. 172 00:15:15,880 --> 00:15:19,090 So this wasn't some idea that everybody else had missed. 173 00:15:19,120 --> 00:15:22,720 He'd taken what other people in the area were doing. 174 00:15:23,470 --> 00:15:29,379 And so just the most extraordinary technical insights and perseverance to kind of push through 175 00:15:29,380 --> 00:15:32,890 an argument that sort of everybody else had ruled out because they knew it wouldn't work. 176 00:15:33,220 --> 00:15:38,800 It's just extraordinary. And everybody is super excited. And the experts in the area checked the proof and yep, it was all good. 177 00:15:40,110 --> 00:15:44,580 And then people started to think, well, can we improve on that? In order to tell you a bit more about that, 178 00:15:44,590 --> 00:15:50,910 I need to tell you a little bit more about finding other kinds of patterns in the primes, not just pairs of twin primes. 179 00:15:51,480 --> 00:15:54,510 So I made myself a number line. 180 00:16:00,070 --> 00:16:04,030 Here's my number line and I call it the primes a different colour. 181 00:16:04,050 --> 00:16:08,470 So I hope you can just about see that the primes are in red there and the non primes are in black. 182 00:16:09,370 --> 00:16:13,060 And then I make myself a twin prime detector. 183 00:16:13,180 --> 00:16:17,680 So it's a piece of card and I've just cut out holes. It's a little bit like an old fashioned punch card. 184 00:16:17,920 --> 00:16:22,630 And the idea is that you can slide it along the number line and detect pairs of twin primes. 185 00:16:23,200 --> 00:16:27,490 So one of the three is no good, is not prime two or four not so good. 186 00:16:27,760 --> 00:16:32,049 Three or five twin primes. Great. Four, six, no, five and seven. 187 00:16:32,050 --> 00:16:35,200 So so I can take my piece of cards, slide it to the number line. 188 00:16:35,350 --> 00:16:42,580 And the twin primes conjecture predicts that there are infinitely many positions of the card where both of the visible numbers are prime. 189 00:16:43,380 --> 00:16:43,500 Right. 190 00:16:43,660 --> 00:16:52,810 So the twin primes conjecture says as you slide the cards along, there are infinitely many pairs of positions where both of these numbers are prime. 191 00:16:53,480 --> 00:16:57,880 And when you start looking at it that way, you start thinking, Well, what if I use other kinds of punch card instead? 192 00:17:01,070 --> 00:17:07,660 Here's one I made earlier. So this one's got three holes and I've got a slightly larger gap. 193 00:17:08,100 --> 00:17:12,740 And again, I can slide this along and see what happens. So no. 194 00:17:12,750 --> 00:17:16,380 Two, four, seven. No good. Three, five, eight. No. 195 00:17:17,190 --> 00:17:22,240 Four, six, nine. No. Five, seven, ten, no. 196 00:17:22,250 --> 00:17:27,610 So again, the goal is to size it alongside infinitely many positions where all of the visible numbers are prime. 197 00:17:28,240 --> 00:17:33,610 So far I have found no positions of the card where all of the visible numbers are prime. 198 00:17:34,560 --> 00:17:38,110 And actually, if you look carefully, we can sort of see what the problem is here. 199 00:17:38,150 --> 00:17:45,210 So if you look at the two numbers on the right. One of them's out, and one of them's even always. 200 00:17:45,310 --> 00:17:50,510 So they differ by three. So if I start with order, if I get the other one afterwards. 201 00:17:50,520 --> 00:17:53,520 So the problem is we know two is the uneven prime. 202 00:17:53,790 --> 00:17:59,820 So no matter how I slide this along, I'm always getting one even one odd, I'm not getting primes. 203 00:18:00,240 --> 00:18:01,890 So this was a very bad choice of card, 204 00:18:03,660 --> 00:18:11,430 but I could try a more sensible choice of card where I fix it so that all of the visible numbers can be odd simultaneously. 205 00:18:11,760 --> 00:18:14,940 Like three, five, seven. In fact, here all of the visible numbers are prime. 206 00:18:14,950 --> 00:18:18,509 So this is great. This is a good triple from our point of view, and then I can slide it along. 207 00:18:18,510 --> 00:18:22,890 And there's really no point in focusing on even numbers because we are not interested in even numbers bigger than two. 208 00:18:23,430 --> 00:18:26,640 So five, seven, nine close. We're not quite. 209 00:18:27,710 --> 00:18:31,400 Seven, nine, 11. Okay, nine is a problem here, so let's keep going. 210 00:18:31,430 --> 00:18:35,390 11, 13, 15. No, now 15 is a problem. 211 00:18:35,430 --> 00:18:39,170 Okay, so I found one triple so far where all of the visible numbers are prime. 212 00:18:39,650 --> 00:18:42,050 But I'm sliding it along and I'm not having much joy. 213 00:18:43,160 --> 00:18:47,900 So what's going on is not the problem of having all even because all of the visible numbers are odd. 214 00:18:49,110 --> 00:18:52,230 Actually, the problem turns out to be multiples of three. 215 00:18:52,250 --> 00:18:56,450 So the problem, if you look at the problematic numbers, they were nine and 15. 216 00:18:56,450 --> 00:19:04,640 And it turns out if we keep going, that that keeps being the problem. So why do we always get a multiple of three that's visible? 217 00:19:05,600 --> 00:19:14,780 I like to think about this in terms of bars of chocolate. So every number in the world is a multiple of three or one. 218 00:19:14,780 --> 00:19:19,430 More than a multiple of three or two more than a multiple of three as illustrated by bars of chocolate. 219 00:19:20,630 --> 00:19:25,970 So every number is one of these. And if we look back at those triples. 220 00:19:26,780 --> 00:19:33,259 One of the visible numbers is of each type. So looking at the bottom 15 is a multiple of three, 13 is one more than a multiple of three, 221 00:19:33,260 --> 00:19:40,280 11 is two more than multiple of 35795 is two more than multiple 371 more nine is a multiple of three. 222 00:19:40,880 --> 00:19:47,570 We always seem to get one of each type and if you think about it, you can think about it with a box of chocolate. 223 00:19:47,570 --> 00:19:52,879 That's sort of inevitable. If you start with a multiple of three, that's bad. If you start this a number, there's one more than a multiple of three. 224 00:19:52,880 --> 00:19:55,640 When you add two one, you get a multiple of three and so on. 225 00:19:55,640 --> 00:20:03,890 So, so what we can see by thinking about bars of chocolate or by some other way is that we always have one of each type of number, 226 00:20:04,820 --> 00:20:08,570 and that means that we somehow deemed always to have a multiple of three that we can see. 227 00:20:08,840 --> 00:20:15,200 And the problem is the only prime multiple of three is three. So in fact it was the same kind of thing with the odds. 228 00:20:15,200 --> 00:20:19,759 And even it's just that we talk about ordered even relative to being a multiple of two or one more than multiple of two, 229 00:20:19,760 --> 00:20:21,020 but it's the same kind of thing. 230 00:20:21,500 --> 00:20:28,470 So this card we can see we're definitely not going to have infinitely many positions of the card where all of the visible numbers are prime. 231 00:20:28,490 --> 00:20:32,930 In fact, the only one that works is three, five and seven. You could kind of convince yourselves of that. 232 00:20:33,920 --> 00:20:40,999 And you can imagine if I had a code with more holes that I might find that the problem was the prime 11 So there are 11 types of number, 233 00:20:41,000 --> 00:20:42,500 like 11 bars of chocolate. 234 00:20:42,500 --> 00:20:47,590 If you think about divisible by 11, you could be a multiple of 11 or one more than a multiple of 11 or two more than a multiple of 11. 235 00:20:47,590 --> 00:20:49,910 And so of the Big Ten, more than a multiple of 11, 236 00:20:50,570 --> 00:20:56,870 if your punch card somehow means that the visible numbers between them always include one of each type, 237 00:20:57,740 --> 00:21:02,000 you are always going to have a multiple of 11 and the only prime multiple of 11 is 11. 238 00:21:02,000 --> 00:21:03,230 So that's going to be no good. 239 00:21:03,740 --> 00:21:11,390 So if our game is think of punch cards where there are infinitely many positions of the card for which all of the visible numbers are prime, 240 00:21:11,660 --> 00:21:15,139 we need to somehow rule those ones out. And mathematicians have a definition for this. 241 00:21:15,140 --> 00:21:20,960 So we talk about a punch card being admissible if there's no problematic prime. 242 00:21:21,410 --> 00:21:25,219 So we've seen a couple of examples of punch cards that are not admissible. 243 00:21:25,220 --> 00:21:32,690 So the first one was this 11 1316 here where we always had an order and even the problem was the prime to the nine, 244 00:21:32,690 --> 00:21:35,840 11, 13, the one I'm holding here, the problematic prime was three. 245 00:21:36,910 --> 00:21:42,879 So not admissible punch cards we can rule out immediately from our game because somehow they are not going to work. 246 00:21:42,880 --> 00:21:47,840 We can see straightaway they're not going to work. The admissible punch cards. 247 00:21:48,860 --> 00:21:52,100 We can't rule them out, but we don't know for certain that they do work. 248 00:21:52,110 --> 00:21:56,030 So the first punch card I've got here is just the twin primes detector. 249 00:21:56,540 --> 00:22:01,100 So we think there are infinitely many positions of the card for which both these visible numbers are prime. 250 00:22:01,370 --> 00:22:04,720 We can't yet prove that. I've got another example of a visible card here. 251 00:22:04,730 --> 00:22:09,920 So you can you can check, but there aren't any primes that are going to cause this card problems. 252 00:22:11,480 --> 00:22:13,290 All that means is that we can't rule it out. 253 00:22:13,310 --> 00:22:19,820 We don't yet know that there are infinitely many positions of the card for which all of the visible numbers are prime. 254 00:22:20,060 --> 00:22:23,180 So that was one of those things where I thought more if you didn't take in the details. 255 00:22:23,180 --> 00:22:28,940 What we need to know is that admissible punch cards all good is still a work in progress. 256 00:22:29,330 --> 00:22:34,040 So Zhang's work used this idea of admissible, admissible punch cards. 257 00:22:34,580 --> 00:22:38,390 They're not called punch cards in the mathematical literature they could set so admissible sets. 258 00:22:38,810 --> 00:22:47,090 And what Zhang proved was that if you've got an admissible punch card with enough holes applied that way, that means loads. 259 00:22:47,330 --> 00:22:49,370 So you've got a punch card with loads of holes. 260 00:22:50,240 --> 00:22:56,270 There are infinitely many positions of that punch card where at least two of the visible numbers are prime. 261 00:22:57,210 --> 00:23:00,570 So we'd like to have all of the visible numbers, but that's way too much to hope for. 262 00:23:00,810 --> 00:23:03,540 But at least two of the visible numbers will be prime. 263 00:23:04,110 --> 00:23:12,150 And then, crucially, there is an invisible punch card that has enough holes where the length of the card is 70 million. 264 00:23:12,600 --> 00:23:16,200 So as I take this card of length, 70 million is sliding along the number line. 265 00:23:16,470 --> 00:23:23,100 Zhang's theorem says there are infinitely many positions where at least two of those visible numbers are both prime. 266 00:23:23,520 --> 00:23:26,730 It might be lot more than that. We think it is. There's one that we know at least two. 267 00:23:26,970 --> 00:23:30,800 So it might be these two over here and these two when we've moved the card further down. 268 00:23:30,810 --> 00:23:36,240 So that's why we don't know exactly what the gap is. All we know is that some two of these visible numbers would be prime. 269 00:23:37,990 --> 00:23:39,490 What do you think about the theorem in that way? 270 00:23:39,970 --> 00:23:45,129 Just start having opportunities to start improving on this, because when people started reading Zhang's paper, 271 00:23:45,130 --> 00:23:49,240 it became clear 70 million was not the best answer that his argument would give. 272 00:23:49,810 --> 00:23:53,740 And you can totally see why that is from his point of view, because from Zhang's point of view, 273 00:23:53,980 --> 00:23:57,610 it doesn't matter whether his paper says 70 million or 65 million. 274 00:23:57,640 --> 00:24:03,790 Right. It really doesn't matter. What matters is being the first to get your paper out and writing a paper that is 275 00:24:03,790 --> 00:24:06,850 clear enough and comprehensible enough that other people are going to read it. 276 00:24:06,880 --> 00:24:13,870 So he focussed on writing this very clear paper rather than spending extra time trying to see, well, what's the best possible number. 277 00:24:13,870 --> 00:24:20,470 But then other people will to go, what, what? What can I do? Because if I could use a narrower card, they still had enough holes. 278 00:24:21,070 --> 00:24:22,660 So we get a better balance straightaway. 279 00:24:22,930 --> 00:24:28,570 If we could somehow show that we could get away with having fewer holes in our punch cards, that we could have a narrower punch card and we'd improve. 280 00:24:28,870 --> 00:24:32,470 So people started trying to improve with this. So this really started in public. 281 00:24:32,480 --> 00:24:39,550 So a couple of people started writing papers, but it really took off when Scott Morrison wrote a blog post called I Can't Just Can't Resist. 282 00:24:39,730 --> 00:24:44,590 There are infinitely many pairs of primes at most. 59,470,640 apart. 283 00:24:46,490 --> 00:24:50,600 And he explained how he'd slightly tweaked Zhang's arguments to come up with this number. 284 00:24:51,030 --> 00:24:56,149 Of course, everybody else then wanted to have their moments of having the world record for this problem. 285 00:24:56,150 --> 00:25:00,020 Right. This is one of the most famous also of problems of the whole of mathematics. 286 00:25:00,020 --> 00:25:03,080 If you have an opportunity to hold the world record for 5 minutes, wouldn't you? 287 00:25:03,830 --> 00:25:07,590 So people started commenting on Scott Morrison's blog post and say, Well, what if we do this? 288 00:25:07,610 --> 00:25:11,240 What if we do that? Maybe we should get this other thing. Somebody which paper a while ago. I think that might help. 289 00:25:12,080 --> 00:25:17,480 And this was sort of going to get out of hand because how do you have that kind of mathematical conversation in public? 290 00:25:17,510 --> 00:25:23,000 You sort of have to be a little bit organised about it. Fortunately, there was a way to organise that conversation. 291 00:25:23,540 --> 00:25:29,419 So a few years earlier, a mathematician called Tim Gowers, who has a blog that's very, 292 00:25:29,420 --> 00:25:34,040 very widely read amongst mathematicians and those beyond the academic maths community. 293 00:25:34,220 --> 00:25:38,720 So he wrote a blog post called Is Massively Collaborative Mathematics Possible? 294 00:25:39,320 --> 00:25:45,680 And he had this kind of vision for a collaborative project on a much larger scale than is usual in maths, in maths. 295 00:25:45,920 --> 00:25:50,120 It's kind of unusual to have a paper with five authors on it, even in pure masses sort of. 296 00:25:50,360 --> 00:25:52,580 There's lots of small groups, but not these big groups. 297 00:25:53,150 --> 00:25:59,000 Tim Gowers wanted to know could we have a much bigger group where somehow everybody works on their little bits of the project, 298 00:25:59,000 --> 00:26:04,729 but no one person has to think too hard. So sort of a little bit like an A.A. Everybody's doing their own thing. 299 00:26:04,730 --> 00:26:06,230 The collective effort is great. 300 00:26:06,470 --> 00:26:14,240 No one person is organising it, and he puts out this post and describes some rules for how this would work, how you should behave in the conversation. 301 00:26:14,480 --> 00:26:18,350 If this led to a publication, who would get the credit? All of those kinds of things. 302 00:26:18,350 --> 00:26:24,400 So he thought through that very carefully and he proposed a project for Polymath to work on. 303 00:26:24,410 --> 00:26:25,970 So there's a joke in that name somewhere. 304 00:26:26,330 --> 00:26:31,190 So the idea was Polymath is going to work on this particular research form, which was a genuine research problem. 305 00:26:31,190 --> 00:26:35,030 People in the area had wanted to solve this problem for some time, have thought hard about it. 306 00:26:35,880 --> 00:26:43,380 And I think to everybody's surprise, within a few weeks, Polymath had proved this result. 307 00:26:43,380 --> 00:26:48,960 They found a new way of proving this conjecture. They'd solved the problem, which was kind of unexpected. 308 00:26:50,160 --> 00:26:53,280 And it seems that that collaboration proved very effective for doing that. 309 00:26:53,280 --> 00:26:58,439 So that led on to other Polymath projects trying out different kinds of problem. 310 00:26:58,440 --> 00:27:05,400 And one of the things here is this very new way of working. If this is all happening in public, on blogs, on wikis, anybody can join in. 311 00:27:05,640 --> 00:27:09,990 It's all in public. So what kind of problems lend themselves to that way of working? 312 00:27:09,990 --> 00:27:14,280 So there's a lot of trying this out. Over the years, some projects were more successful than others, 313 00:27:14,820 --> 00:27:19,950 but it meant it was very natural when this came along to say, Well, let's have a Polymath project to work on this. 314 00:27:20,430 --> 00:27:27,299 So Terry Tao, who is another mathematician with a very widely read blog who'd been involved in the earlier Polymath project, 315 00:27:27,300 --> 00:27:31,500 said, Let's have Polymath eight projects on bounded gaps between primes. 316 00:27:31,740 --> 00:27:38,310 And I've included this quote This is his description, his proposal for what the project should do, because I think it's quite interesting. 317 00:27:38,490 --> 00:27:41,610 Yes, the first one is improving those numerical bounds. 318 00:27:41,610 --> 00:27:44,670 We've got 70 million. We've got Scott Morrison's bounce. How far can we go? 319 00:27:45,030 --> 00:27:52,499 But the second aspect to the problems of the project is understanding what sounded did, really kind of getting to grips with the details. 320 00:27:52,500 --> 00:27:56,940 What were his new insights that other people hadn't had before? Could those be simplified? 321 00:27:56,940 --> 00:28:00,990 Could they be streamlined? How did it fit with what other people in the area had done? 322 00:28:01,170 --> 00:28:07,319 Mathematics doesn't happen in isolation. Yes, John worked by himself on this, but he built on what other people had done. 323 00:28:07,320 --> 00:28:09,930 It was going to feed into other things. So what were those relationships? 324 00:28:10,350 --> 00:28:17,969 So this Polymath project started and of course lots of people were quite keen to join in because their opportunity to be part of the whole thing, 325 00:28:17,970 --> 00:28:24,450 the world record on this very famous problem and it's all in public, it's all in blogs, it's all online, you can still find it. 326 00:28:24,750 --> 00:28:30,570 I quote you if you're interested to have a look. So what I've done here is taken some screenshots from the wiki that was set up for the project. 327 00:28:30,960 --> 00:28:35,780 This is like a league table for the best known things, so I know it's hard to see on the screen. 328 00:28:35,790 --> 00:28:39,209 I'm not expecting you to kind of see all the details. The crucial call and this is comments. 329 00:28:39,210 --> 00:28:43,110 The crucial column is the next one across, which is it's called H at the top. 330 00:28:43,110 --> 00:28:44,939 That's the crucial number that we're interested in. 331 00:28:44,940 --> 00:28:52,590 So the second row here is John with H 70 million, infinitely many pairs of primes that differ by at most 70 million. 332 00:28:53,250 --> 00:28:54,600 And then this number tumbled. 333 00:28:54,600 --> 00:29:01,169 So Scott Morrison is quite high up on that list, so fairly early and then this number just tumbles through the summer of 2013. 334 00:29:01,170 --> 00:29:06,149 So we've got 2nd June 2013 already. The number of digits is decreasing significantly. 335 00:29:06,150 --> 00:29:11,270 Even if you call me the number, they're getting much smaller. And it kept going. 336 00:29:12,020 --> 00:29:14,320 Actually, this was quite an interesting moment for the league table. 337 00:29:14,330 --> 00:29:19,370 I don't know whether you can see lots of these numbers have been crossed out because what would happen when people come along and say, 338 00:29:19,370 --> 00:29:20,810 well, I think I could get this number. 339 00:29:21,110 --> 00:29:28,189 Here's a link to the blog or wherever that I've explained the argument, and then other people will come along and check and go, Yep, that seems great. 340 00:29:28,190 --> 00:29:35,890 Or Hang on, I don't understand. This bit seems to be a problem there. There was a whole sequence of results here that turned out to be problematic. 341 00:29:35,950 --> 00:29:43,450 And they were all based on a preprint, like a draft paper by one of the experts in the area, Jonas Prince, who put up this draft paper. 342 00:29:43,660 --> 00:29:47,110 And people said, Oh, that's great using these ideas. Here's how we can improve the bounds. 343 00:29:47,920 --> 00:29:52,180 And it turned out there was a mistake in the draft. So these bans had to be retracted. 344 00:29:52,690 --> 00:29:58,510 I see that the paper went on to be interesting and useful and have consequences, but not quite for these numerical bounds. 345 00:29:59,020 --> 00:30:03,730 And I think one of the really interesting things about this polymath way of working, so we get to see all of this. 346 00:30:04,360 --> 00:30:10,560 Normally when mathematicians work on a problem, they write up their nicely polished solution at the end for publication. 347 00:30:10,570 --> 00:30:15,490 What appears in the journal, what appears in the paper online is the This is how I did it. 348 00:30:15,490 --> 00:30:21,459 This is what works. This is all cleaned up. You don't see all the rest of the What if we try this? 349 00:30:21,460 --> 00:30:25,090 What if we try that? Oops, that thing didn't work out. I got that bit wrong. 350 00:30:25,090 --> 00:30:28,180 All of those other kinds of things. They get wiped off the boards at the end of the project. 351 00:30:28,180 --> 00:30:32,200 They get put in the recycling bin or put in a pile in the office to look at later. 352 00:30:33,100 --> 00:30:35,920 Gauss, the great number theorists of the past, 353 00:30:36,970 --> 00:30:41,500 commented that the architects don't leave the scaffolding behind and this is somehow what mathematicians do. 354 00:30:41,860 --> 00:30:46,689 But I see this quite frustrating because it means if you've got some famous problem that everybody's working well, 355 00:30:46,690 --> 00:30:49,660 maybe there are three obvious things to try to solve this problem, 356 00:30:49,960 --> 00:30:56,080 and everybody who works in this problem tries those three obvious things before they get on to maybe having some new ideas, 357 00:30:56,530 --> 00:31:02,379 having something like this. There's a record of everything that you've tried so other people can see, Oh, I see. 358 00:31:02,380 --> 00:31:06,610 That seems to be a bit of a dead end or that looks promising, but they didn't pursue that. 359 00:31:06,610 --> 00:31:11,139 Maybe I can use that. The good news is that after those patterns were retracted, 360 00:31:11,140 --> 00:31:15,910 there was then a continued improvement and it just dropped and dropped through the summer 2013. 361 00:31:16,330 --> 00:31:23,530 I'm sure that that progress would have happened without Polymath, but Polymath was a very good way of streamlining it and making it more efficient. 362 00:31:23,530 --> 00:31:28,569 Otherwise you'd have had maybe individuals, maybe some groups of collaborators putting out papers saying, 363 00:31:28,570 --> 00:31:32,049 Well, I can get this band, I can get this band, I can get this band, this kind of race. 364 00:31:32,050 --> 00:31:35,020 So doing it collectively was much more efficient way of doing it. 365 00:31:35,410 --> 00:31:41,790 And through the course of doing it, it became clear that the different ways to improve on the band drew on different types of expertise. 366 00:31:41,800 --> 00:31:49,360 So some people were very good at getting their computers to do a particular kind of calculation to find narrow punch cards with lots of holes. 367 00:31:49,660 --> 00:31:54,040 Other people were more interested in the theoretical aspects of How could we decrease the number of holes? 368 00:31:54,040 --> 00:32:01,560 How can we really understand this argument? So in order to go forward with the story, I want to go back a bit to before then. 369 00:32:02,430 --> 00:32:09,149 So back in 2005, when there was the previous big breakthrough on this problem, which came from Goldstone Pinson, 370 00:32:09,150 --> 00:32:15,270 Yildirim In fact the same points I just mentioned and Goldstone Pitts and Yildirim proved that if the Halberstam conjecture is true, 371 00:32:16,170 --> 00:32:19,830 then there are infinitely many pairs of crimes that differ by, at most, 16. 372 00:32:20,790 --> 00:32:27,460 That's like 16. The trouble is, we can't prove the Halberstam conjecture. 373 00:32:28,540 --> 00:32:36,879 So this is a feel slightly disappointing. At the time. This was super exciting because it was progress on a problem that had seemed intractable. 374 00:32:36,880 --> 00:32:41,530 They'd found a way to do something, make some progress, and actually assuming a conjecture, 375 00:32:41,530 --> 00:32:44,590 you don't have to prove it's a really good way of making progress in maths. 376 00:32:44,980 --> 00:32:53,050 Sometimes if you can isolate that the problem is here and you say, Well, if I could, if I'll allow myself to assume that and deduce this from it, 377 00:32:53,470 --> 00:32:56,740 that could be a really helpful way of focusing attention on what's the important thing here. 378 00:32:56,770 --> 00:33:03,070 What are the key questions we need to do? So in number theory, often the the assumption you want to make is called the Riemann hypothesis. 379 00:33:03,100 --> 00:33:06,640 This is another of these famous unsolved problems. So there are lots of papers in mathematics and. 380 00:33:07,150 --> 00:33:10,690 If the Riemann hypothesis is true, then whatever. 381 00:33:12,280 --> 00:33:18,430 In the early part of the 20th century, a group of mathematicians proved if the Riemann hypothesis is true, then something or other is true. 382 00:33:18,880 --> 00:33:24,520 And then a few years later, another group came along and proof. If the Riemann hypothesis is false, that same statement is true. 383 00:33:26,560 --> 00:33:29,380 Isn't that beautiful? We know it's true. We just know. Oh, it's proof. 384 00:33:30,730 --> 00:33:35,260 The go to conjecture in this particular area is called the Elliott Halberstam conjecture, 385 00:33:35,260 --> 00:33:38,470 which is a conjecture about the distribution of prime numbers. 386 00:33:39,490 --> 00:33:41,229 So let me try to give you a little bit of a flavour. 387 00:33:41,230 --> 00:33:45,010 I don't want to use the technicalities, but let me try to give you a flavour of the Halberstam conjecture. 388 00:33:45,310 --> 00:33:50,980 So what I've got here is the prime numbers, but instead of ten columns like I had earlier this time, I've got six columns. 389 00:33:52,130 --> 00:33:55,520 I think this is quite visually striking. Patterns start to leap out. 390 00:33:55,970 --> 00:34:01,550 So the four column and the six column. Appear not to have any primes. 391 00:34:01,760 --> 00:34:05,749 Does that continue if I go below the slide? Yes. Because of the numbers in the fourth column. 392 00:34:05,750 --> 00:34:10,530 In the sixth column. Even when I choose the uneven prime. So if two is very lonely in its column. 393 00:34:10,790 --> 00:34:14,180 Two is the only even prime, no matter how far down we go. Don't get any more there. 394 00:34:15,650 --> 00:34:21,650 Three. He's looking pretty lonely, too. But we know that the only multiple of three that's prime is three. 395 00:34:21,800 --> 00:34:25,550 And if you look carefully at this column, these numbers are all multiples of three. 396 00:34:26,180 --> 00:34:29,329 So these are the multiples of six, the even multiples of three. 397 00:34:29,330 --> 00:34:37,940 These are the odd multiples of three. So what this slide and that little bit of reasoning shows is that apart from two and three, 398 00:34:38,660 --> 00:34:42,560 every prime in the world is one list, less than a multiple of six. 399 00:34:43,930 --> 00:34:47,400 Or one more than a multiple six. Isn't that nice? 400 00:34:48,090 --> 00:34:52,620 It's like, good. I'm going to say it again. Apart from two and three. 401 00:34:53,220 --> 00:34:56,730 Every problem in the world is one. Less than a multiple of six. 402 00:34:57,590 --> 00:35:03,770 Over more than a multiple of six. And once you know that, then you can start asking more questions about the distribution of the primes. 403 00:35:03,920 --> 00:35:07,910 Like we know they're infinitely many primes. We saw Euclid's proof of that earlier on, 404 00:35:08,450 --> 00:35:12,739 other infinitely many that are one less than a multiple six other infinitely many that are one more than a multiple of six. 405 00:35:12,740 --> 00:35:18,240 There must be infinitely many. At least one of those two. But is it just one or is it both? 406 00:35:18,390 --> 00:35:22,260 So it turns out there are infinitely many primes in both of those columns. 407 00:35:22,890 --> 00:35:28,110 So this one turns out you can prove quite nicely by adapting Euclid's argument. 408 00:35:28,320 --> 00:35:34,010 This one takes a little bit more work, and you can ask other questions, like if you were a prime number. 409 00:35:35,010 --> 00:35:38,190 Would you rather be one less than a multiple of six or one more than a multiple of six? 410 00:35:39,630 --> 00:35:44,520 So I guess the more mathematical way to say that is, if we look at the numbers, up to a million or a billion or whatever. 411 00:35:44,790 --> 00:35:48,960 Are there more problems that are one less than multiple six or more primes that are one more than a multiple of six? 412 00:35:49,830 --> 00:35:54,210 And it's not really clear that the primes should favour one of those options over the other. 413 00:35:54,220 --> 00:35:57,600 So we kind of expect the to be pretty evenly distributed. 414 00:35:57,600 --> 00:36:01,740 So I think up to 90 here, they are pretty evenly divided between those two. 415 00:36:01,950 --> 00:36:08,939 So don't expect to be exactly even, but pretty close. So the Halberstam conjecture is a prediction that, yes, 416 00:36:08,940 --> 00:36:13,320 they should be they should be about evenly divided between those two columns 417 00:36:13,800 --> 00:36:18,740 and a whole bunch of other similar predictions for other numbers of columns. 418 00:36:18,750 --> 00:36:24,569 So if I've got seven columns or 101 columns, whatever, I can predict which columns should have infinitely many primes. 419 00:36:24,570 --> 00:36:30,960 In fact, we can prove that that turns out to be a harder theorem to prove pretty interesting theorem which columns have infinitely many primes. 420 00:36:31,440 --> 00:36:37,830 And the Halberstam conjecture says that the Prime should be pretty evenly distributed between the columns, whether infinitely many primes. 421 00:36:38,070 --> 00:36:43,830 So it's a for all of these tables at once. So that turns out to be hard. 422 00:36:44,460 --> 00:36:51,450 By which I mean we don't currently know how to prove it. But if you take that data, that kind of prediction about the distribution of the primes, 423 00:36:52,020 --> 00:36:57,300 you can feed it in to this argument using sieve theory, which is what Goldstein yields from did. 424 00:36:57,690 --> 00:37:00,870 And they were able to say, well, if this is true, then we can prove this result. 425 00:37:02,250 --> 00:37:06,149 So there's kind of a whole family of these conjectures, the different, different parameters, 426 00:37:06,150 --> 00:37:08,820 and they sort of needed a particular parameter where they couldn't prove it. 427 00:37:10,410 --> 00:37:19,350 What Zang was able to do was to slightly weaken the assumption you needed to make so something slightly weaker than the hope of some conjecture. 428 00:37:19,920 --> 00:37:24,330 Increasingly, then, he could prove that required facts about the distribution of the Prime. 429 00:37:24,340 --> 00:37:31,740 So he took the goals of Pennsyl Grimm kind of idea and then adapted it in a really ingenious way to make it work out. 430 00:37:32,520 --> 00:37:33,750 So back to Polymath. 431 00:37:34,350 --> 00:37:40,740 So by August 2013, the Polymath Project had managed to prove that there are infinitely many paths of primes that differ by at most. 432 00:37:43,100 --> 00:37:48,600 4680. I love the fact that some of you are looking at press. 433 00:37:48,780 --> 00:37:52,069 If I say 4680 half an hour, you'd have said That's rubbish. 434 00:37:52,070 --> 00:37:56,570 She wants it to be. But it is a big improvement on 70 million. 435 00:37:56,650 --> 00:38:06,830 Right. It's fantastic. Very efficient. The trouble is, by August 2013, so totally said that the low and medium hanging fruit is being picked. 436 00:38:07,070 --> 00:38:11,690 All those little tweaks you could do to Zhang's argument or those getting a computer to help us a bit, 437 00:38:12,050 --> 00:38:16,220 that sort of apparently will be exhausted and it seemed like it was going to need to be a new idea. 438 00:38:17,030 --> 00:38:21,740 The nature of research, of course, is you never know when the next idea is going to come or where it might come from. 439 00:38:22,100 --> 00:38:26,420 In this case, it didn't take very long. In fact, it was November 2013, 440 00:38:27,770 --> 00:38:34,670 and a mathematician called James Maynard was able to show that there are infinitely many pairs of primes that differ by, at most, 600. 441 00:38:35,000 --> 00:38:41,330 Big improvement, James said recently finished his Ph.D. He's now one of our colleagues in Oxford. 442 00:38:41,540 --> 00:38:48,829 He'd gone to Montreal as a postdoc at the time, and through his Ph.D. work, he'd become an expert in this kind of area. 443 00:38:48,830 --> 00:38:54,470 And he found that he was able to go back to the work of Goldstein Points Yildirim, and find a new way through. 444 00:38:54,800 --> 00:39:01,880 So not only did his argument give a better number than the work of Zhang, a polymath, but it also gave a slightly simpler argument. 445 00:39:02,210 --> 00:39:08,900 And that's the kind of dream scenario, right, where a slightly easier argument gets you a better bound sort of on the right track with understanding. 446 00:39:11,520 --> 00:39:17,100 One of the nice things about this pound is that it's small enough that I can show you an invisible set that works. 447 00:39:17,310 --> 00:39:23,040 I didn't expect you to take in these numbers. This is like the the recipe for the punch card, the admissible punch card. 448 00:39:23,050 --> 00:39:27,710 So there are infinitely many positions of this punch card where at least two of the visible numbers apply. 449 00:39:27,870 --> 00:39:32,820 And I'm only putting this up to make the point that it's a small enough set so I can fit it on the screen. 450 00:39:34,250 --> 00:39:39,320 So James Menon could say that infinitely many pairs of poems that differ by at most, 600. 451 00:39:39,650 --> 00:39:44,180 Of course, a polymath gets very excited and said, Well, that's great. What if we look at James Maynard's? 452 00:39:44,180 --> 00:39:48,440 Well, I can understand that. Can we do even better? So this league table resumes. 453 00:39:48,440 --> 00:39:52,219 The pound kept dropping and dropping by April 2014. 454 00:39:52,220 --> 00:39:58,790 So a few months later. Polymath can show that infinitely many pairs of primes that differ by, at most, 246. 455 00:40:01,110 --> 00:40:04,800 And to the best of my knowledge, that's the state of the art. 456 00:40:06,890 --> 00:40:10,700 You'll notice that April 2014 was a little while ago. So. 457 00:40:10,700 --> 00:40:17,089 So Polymaths have written up their paper types may not just written up his paper, I should say about the same time as James Maynard had his ideas, 458 00:40:17,090 --> 00:40:23,540 Terry Tao had some very similar ideas, that same kind of approach for adopting the theoretic argument. 459 00:40:23,870 --> 00:40:27,290 But Toto's argument escapes the slightly less good numerical bounds. 460 00:40:27,650 --> 00:40:34,610 What James Manos and Terry Tao both managed to do was find an argument that would deal with other clusters of primes as well. 461 00:40:34,610 --> 00:40:38,569 So not just pairs of primes, which the differ by two, but if you want three primes, 462 00:40:38,570 --> 00:40:43,970 very close together for primes close together, they started to be able to make progress on those kinds of questions. 463 00:40:44,720 --> 00:40:47,720 So Polymaths have been working on improving those spans too. 464 00:40:47,720 --> 00:40:51,320 And if you go and look at the league table, you kind of track down those bands. 465 00:40:52,700 --> 00:40:58,910 There's still the game of if we're allowed to assume the Halberstam conjecture, what's the best band we can get? 466 00:40:59,420 --> 00:41:02,970 So Goldstein Pitts deals to him back in 2005. 467 00:41:03,110 --> 00:41:07,850 So if you see in the Halberstam conjecture, then there are infinitely many pairs of primes differed by, at most, 16. 468 00:41:08,450 --> 00:41:13,460 So James Maynard's argument gets you down to 12. 469 00:41:13,640 --> 00:41:17,630 So if you see the Halberstam conjecture, there are infinitely many pairs of pipes differ by at most, 12. 470 00:41:18,110 --> 00:41:22,160 If you assume a stronger and more controversial form, it's the conjecture. 471 00:41:22,550 --> 00:41:28,210 Maybe you could even get all the way down to six. That's like tantalisingly close to two, right? 472 00:41:29,020 --> 00:41:37,299 So near and yet so far. And the trouble is that within Sift theory, there's a well known kind of problem. 473 00:41:37,300 --> 00:41:44,260 That means that people don't think that the arguments that get down to six, even assuming this conjecture will get down to two. 474 00:41:44,620 --> 00:41:45,819 It's called the parity problem. 475 00:41:45,820 --> 00:41:52,479 It's because the theory is a very good at dealing with only looking for numbers with an even number, an odd number of prime factors. 476 00:41:52,480 --> 00:41:57,130 This is kind of technical difficulty. That means the six is like a brick wall. 477 00:41:58,030 --> 00:42:01,650 So they're going to need to be some new ideas. So where next? 478 00:42:01,660 --> 00:42:04,830 Well, there's all this work on the ten primes conjecture to look at. 479 00:42:04,840 --> 00:42:06,880 So, of course, people are trying to improve on the bounds. 480 00:42:07,480 --> 00:42:12,520 People are taking the arguments that have been developed and saying, well, what else could we do with those? 481 00:42:12,550 --> 00:42:21,850 What other problems can be solved with these kinds of ideas? Surprisingly, sometimes the answer is problems about large gaps between primes. 482 00:42:22,240 --> 00:42:26,590 So how large a gap can you have between primes where you can have as large a gap as you like? 483 00:42:26,590 --> 00:42:30,549 If you want a gap of 100 between two consecutive primes, you can arrange that. 484 00:42:30,550 --> 00:42:32,379 That's not that's not too difficult. 485 00:42:32,380 --> 00:42:38,140 That's the kind of problem that might appear on an undergraduate problem sheets is a nice go way to think about this kind of thing. 486 00:42:39,230 --> 00:42:42,260 The interesting question is when those gaps are larger than the average. 487 00:42:42,260 --> 00:42:45,910 So on average, the primes get more spread out and the gaps can be as large as we like. 488 00:42:45,920 --> 00:42:52,880 But amongst primes around a million, we sort of know what the average gap looks like, and sometimes the gap can be much smaller, like two. 489 00:42:53,090 --> 00:42:57,320 Sometimes the gap could be much larger. So you sort of say, well, four primes around this size. 490 00:42:57,950 --> 00:43:02,179 Can the gap be much larger than the average? And this was a longstanding problem. 491 00:43:02,180 --> 00:43:05,960 That was a problem posed by Addis, who who was a prolific poser of problems. 492 00:43:06,230 --> 00:43:11,629 And he likes this problem and thought it was so difficult that he attached quite a large monetary value to this problem. 493 00:43:11,630 --> 00:43:14,930 So I'd like to offer cash prizes in return for problems. 494 00:43:15,170 --> 00:43:19,970 And the amount of cash he was willing to stump up depended on how hard he thought the problem would be. 495 00:43:20,810 --> 00:43:31,820 And this died some years ago now, but a friend colleague of his has been honouring these promises subsequently and simultaneously. 496 00:43:32,450 --> 00:43:36,649 Two groups made progress on this. How large can the gap be? 497 00:43:36,650 --> 00:43:38,660 When can it be larger than the average gap problem? 498 00:43:39,050 --> 00:43:44,420 So one of those groups was a group of four mathematicians involving in fact Ben Green here from Oxford, 499 00:43:44,420 --> 00:43:48,140 Terry Tao, couple plus couple of others come forward and say again, 500 00:43:48,500 --> 00:43:53,930 the other group was James Maynard, who they came up with these completely different approaches, 501 00:43:54,650 --> 00:43:58,580 each based on things that those groups were experts in before and what's extraordinary. 502 00:43:58,580 --> 00:44:05,389 So James Maynard took his work on understanding small gaps between primes and used it to understand the large gaps between primes, 503 00:44:05,390 --> 00:44:10,490 which is very unexpected. So there's lots of potential for seeing where these ideas can take us, 504 00:44:10,490 --> 00:44:14,900 even if they're not proof of the two in primes conjecture, they're all building our understanding of prime numbers. 505 00:44:15,200 --> 00:44:20,120 I do believe that the twin primes conjecture is true. I do believe we managed to prove it. 506 00:44:20,810 --> 00:44:24,620 I've got no idea when it might be next week. 507 00:44:24,800 --> 00:44:30,620 It might be next year, it might be in 20 years time. I'm one of the young people in the audience today has solved it. 508 00:44:30,950 --> 00:44:33,620 I'm looking forward to finding out. Thank you very much.