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.