1
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I'd like to start with a question. What exactly is mathematics?

2
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What is it that mathematicians in the building here at the University of Oxford are doing all day up in their offices?

3
00:00:26,940 --> 00:00:33,540
I think most people have some sort of idea of what it is that the physicists and the chemists and the biologists are doing.

4
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Chemists mixing chemicals together, blowing things up, the physicists smashing things together in their collider.

5
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What is it? What is it? All mathematicians do.

6
00:00:43,770 --> 00:00:51,929
I think actually if you ask somebody on the street, most people's perception is that somehow we're doing some mathematics is about multiplication,

7
00:00:51,930 --> 00:01:00,330
percentages, long division and as research mathematicians, well, perhaps we're in long division to a lot of decimal places.

8
00:01:00,480 --> 00:01:05,190
I think perhaps, you know, we'd been put out of a job by a computer now,

9
00:01:05,940 --> 00:01:10,530
but I think that's really a failure of the way that we teach mathematics at school.

10
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I think in some sense we teach mathematics a little bit like we teach a musical instrument as if all

11
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you're allowed to play are scales and arpeggios and you never actually hear a real piece of music.

12
00:01:24,090 --> 00:01:32,640
And so that's what I'm going to have to do, say, in this lecture, is to play you some of the music that really excites me about mathematics.

13
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And I think there's actually quite a lot in common between mathematics and music.

14
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And when people ask me, So what is mathematics?

15
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What is a mathematician? I tend to answer by saying, Well, I think a mathematician is a pattern searcher.

16
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Mathematics is the science of patterns, trying to understand a kind of logic and order in the kind of chaotic world around us all.

17
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A similar to the way that music is distinguished from noise,

18
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from beings that patterns against the kind of chaos that you might hear, which is white noise.

19
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So I would say that mathematics is a science of pattern searching.

20
00:02:08,880 --> 00:02:16,709
So I'm going to start off with a few patterns to try and warm you up mathematically, to explore the idea of try and find some underlying logic.

21
00:02:16,710 --> 00:02:20,550
So rather like those puzzles you probably had at school when you're in school,

22
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I'm going to give you a sequence of numbers, and what I want you to do is to try and find the pattern in these numbers.

23
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So I'm going to start over with a few easy ones.

24
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If you've done a mathematics degree or a study for mathematics, were you not allowed to play on the first two?

25
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Okay. So you can join in a bit later. But I'm not ongoing mathematics.

26
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We will. What's the next number in the sequence? One, three, six, ten, 15, 21, 28.

27
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36. Exactly right.

28
00:02:49,640 --> 00:02:56,140
So these are the triangular numbers. Very simply, what you're doing is adding one plus two plus three or four plus five.

29
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And we call these the triangular numbers because you can view them very graphically.

30
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If you take the number of stones you need to make a triangle,

31
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when you add an extra layer on each time and the number of sevens you'll need is given by this sequence, the triangle numbers.

32
00:03:10,390 --> 00:03:15,250
And we understand these numbers incredibly well. For example, we have a formula which will tell you,

33
00:03:15,490 --> 00:03:21,610
say the hundreds of triangular number on this list without having to add up all the numbers from 1 to 100.

34
00:03:22,030 --> 00:03:27,549
And in fact, you can get it if you put together two of these triangles, you can make a rectangle counting things.

35
00:03:27,550 --> 00:03:32,410
And a rectangle is very easy. So the formula is just half the number of things in that rectangle.

36
00:03:32,960 --> 00:03:36,630
Okay. The next sequence, if you read The Da Vinci Code, you know the last page on this one.

37
00:03:36,630 --> 00:03:41,180
I but what's the next sequence? Three things sequence. What's the next number in this sequence?

38
00:03:41,990 --> 00:03:46,910
34. Exactly. And how do you get that set? I mean, it's 21, but I guess.

39
00:03:47,150 --> 00:03:52,969
So there's a it's a a C series of numbers which grows out of the previous numbers in the sequence.

40
00:03:52,970 --> 00:03:56,240
And these are a very famous set of numbers to the Fibonacci numbers.

41
00:03:56,480 --> 00:04:05,330
So you get a sense, boy, 18, 13 and 21, 21, 34 gives you 55, and these are somehow nature's favourite numbers.

42
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You find them all over the natural world. So for example, you take a flower and you count the number of petals in that flower.

43
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Then invariably it's a number in the film, an orgy sequence, or sometimes you get double the number,

44
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you sort of get two copies of the flower, one on top of each other.

45
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And if it isn't the number in the Fibonacci sequence, that's because the petals falling off your flower,

46
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which isn't how mathematicians get round exceptions.

47
00:04:29,810 --> 00:04:33,980
So we we understand these numbers quite well, although there are still some mysteries about these numbers.

48
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We have a formula to calculate. Calculate the Fibonacci numbers.

49
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You don't have to Internet 170 number. You don't have to add up all the pairs all the way up to the hundredth number.

50
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And it's a rather nice one which involves this magic number two, the golden ratio, which expresses a sort of perfect proportions in all its in nature.

51
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So this is a very famous sequence actually discovered not by Stephen Archie verse,

52
00:04:58,670 --> 00:05:05,810
but by Indian musicians and poets who realise these numbers also count rhythms that you can make.

53
00:05:06,530 --> 00:05:10,130
Okay, what about next sequence? One, two, four, 16, 32.

54
00:05:10,700 --> 00:05:15,170
Well, 32 is one off. So that actually the next number in the sequence is actually 31.

55
00:05:16,040 --> 00:05:21,199
So this is a warning to you that this is why mathematics is rather exciting,

56
00:05:21,200 --> 00:05:27,269
because sometimes you think you know where something's going and then it can take a rather dramatic turn in another direction.

57
00:05:27,270 --> 00:05:32,629
And it's these surprises, I think, which I find so wonderful about mathematics.

58
00:05:32,630 --> 00:05:38,900
So let me explain to you why things was equally good in answer as 30 to 32, you think it's just has a two.

59
00:05:39,110 --> 00:05:43,460
But actually these numbers are describing something called the Circle Division numbers.

60
00:05:43,670 --> 00:05:47,730
So what are these? Well, it only take a circle and I put one point on the circle.

61
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I just have one region. But if I put another point and they join them up, then actually I can get I can divide the circle up into two regions.

62
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So 2.4 divide seven into two regions. Now for the third points, if I join all of those, I get a triangle in the middle and four regions out round now.

63
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So three regions around the outside. So about 1 to 4 regions put forth points on join up all the points.

64
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You get this little envelope in the middle, four triangles and all regions around the outside eight regions.

65
00:06:19,490 --> 00:06:22,820
So you think I started the pattern? Now I know what's going to happen next.

66
00:06:23,030 --> 00:06:26,929
Put another dots on your circle. I'm sure now you've got five points.

67
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Join them all up. You have this pentagram in the Pentagon in the mental count, the number of regions, 16 ranges.

68
00:06:33,170 --> 00:06:37,820
And so by now you're absolutely convinced you know what's going to happen when you add the sixth point.

69
00:06:38,000 --> 00:06:42,560
But here's the surprise it has to which controlling number of regions.

70
00:06:42,830 --> 00:06:52,790
So where do you even try this later? If you put six dots, however you arrange in the maximum number of regions you can get surprisingly only 31.

71
00:06:53,060 --> 00:06:56,060
So it is pairs of two, which is controlling this sequence.

72
00:06:56,240 --> 00:07:00,860
And actually another formula that's slightly more complicated is you have to take the number of dots,

73
00:07:00,860 --> 00:07:06,319
race to the power for calculate this polynomial and this gives you the number of regions you get.

74
00:07:06,320 --> 00:07:14,570
So if I want to put 100 dots, this formula will tell me how how many regions all these I'm so a warning here.

75
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They're all surprise. Now, there is the surprises, I think, which make mathematics such an exciting subject.

76
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It's like a piece of music where suddenly you think you know where it's heading and then the composer takes you on completely different journey.

77
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Okay, so warming you are. But here's a more challenging sequence.

78
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So if you've got a master's degree or a professor of maths here at Oxford, you're allowed to join in on this sequence.

79
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What's the next number in the sequence? Ten, nine, ten, 11, 13, 16.

80
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A little quieter. You thought it was also easy?

81
00:07:49,730 --> 00:07:52,760
Triangular number seven asked. You don't call Lenovo's.

82
00:07:54,470 --> 00:07:57,650
The math professors. There are a few maths professors here. You're allowed to play.

83
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5 to 5 active members. What about the teams in the nine?

84
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A little the beginning, Megan. Well, I must admit, if you do manage to get that, 26 is the next number on the sequence.

85
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I recommend you buy a lottery ticket this coming weekend because they've learned about the lottery ticket numbers for the 28th of September last year.

86
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So another warning hat. So pick your battles carefully.

87
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In mathematics, not everything does have concerns,

88
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although even randomness we have managed to apply mathematics to be able to spot structures even in random sequences like this.

89
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Unfortunately, I don't have a formula for these. If I did, I would be here talking to you now.

90
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I'll be on my tropical island, enjoying myself with some. Okay.

91
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So the last was two, three, five, seven, 11, 13, 17, 19.

92
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23. Exactly. And these are the prime numbers.

93
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Probably the most important numbers in the whole of mathematics.

94
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And these are the individual numbers, a number which I can't abide by anything except itself.

95
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And one. So 23 is the next number that we have wanted to be count up to 29.

96
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Now, I would say of all the numbers in mathematics, these are the most important.

97
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They're the most fundamental numbers yet, as I hope to explain to you,

98
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they are also the most challenging sequence of numbers in the whole of all subjects.

99
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These numbers are so important because they literally build all of our numbers.

100
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If we take a number like 105, is that a final it now?

101
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Clearly it's divisible by five. Exactly. So divided by five, I get down to 21 times five.

102
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21 is that point. Now, that's three times seven, but now three times seven times five.

103
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And these are now individual numbers. These are prime numbers.

104
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These are the primes which build that number 140.

105
00:09:55,970 --> 00:09:59,590
I you can do this with any number. Take your telephone number.

106
00:09:59,600 --> 00:10:04,790
It's either going to be indivisible in prime. And if it is, your lucky is always off to one of those respects.

107
00:10:06,230 --> 00:10:09,680
Or you can divide it and you can keep on dividing, dividing, dividing.

108
00:10:09,680 --> 00:10:11,170
And so you get down into this.

109
00:10:12,260 --> 00:10:20,000
So for me, as a mathematician, these prime numbers are honestly like the building blocks of myself in the absence of arithmetic,

110
00:10:20,300 --> 00:10:24,440
because to me they're a little bit like the absence in the periodic table.

111
00:10:24,680 --> 00:10:31,250
The case study is one of the building of the periodic table, solving puzzles able to date to new elements assigned here.

112
00:10:31,790 --> 00:10:35,270
These are the items which build the whole of the molecular moment.

113
00:10:36,200 --> 00:10:40,790
For me, competition and the primes that in my periodic table.

114
00:10:41,600 --> 00:10:47,810
Yes, you look at the primes, the sequence of numbers, these primes build the subject to looking for patterns.

115
00:10:48,160 --> 00:10:52,170
If I look for patterns on these numbers, there seems to be none.

116
00:10:52,170 --> 00:10:58,700
There are two. I've written the primes as a piece and I really think that primes like myself.

117
00:10:59,570 --> 00:11:02,840
So the whole piece, every time it runs over a prime number.

118
00:11:03,230 --> 00:11:07,970
But this seems to be a subject since I have been drinking a lot of caffeine or need to go into the

119
00:11:08,090 --> 00:11:12,920
cognac department because it seems very hard to predict when the next speech is going to occur.

120
00:11:13,490 --> 00:11:18,380
You go 17, 19, 23. They want to be at 29 and a double be there.

121
00:11:19,280 --> 00:11:26,960
And actually, I think many of us opticians feel that actually there's more in common between the prime numbers and those lottery ticket numbers.

122
00:11:27,240 --> 00:11:30,810
Now, between the primes and the Fibonacci and the triangular numbers,

123
00:11:31,280 --> 00:11:37,400
we don't have some magic formula which helps us to calculate where the next point is going to be.

124
00:11:38,060 --> 00:11:43,220
And for me, these points represents one of the biggest challenge for the patent search.

125
00:11:44,600 --> 00:11:48,560
Now, who do you think were the first to actually discover the primes?

126
00:11:48,980 --> 00:11:55,350
And, you know, it is the first to discover primes. The ancient Greeks.

127
00:11:55,560 --> 00:11:59,460
They certainly did some fantastic stuff. But no, they were the first to actually.

128
00:12:00,280 --> 00:12:03,650
Egyptian Egyptians. They were one of the first to do mathematics.

129
00:12:04,420 --> 00:12:11,030
They calculated PI, for example, but no, they were the first to say which one it was.

130
00:12:11,380 --> 00:12:17,740
So now that's where they also did some great math. But at this point in time, when they were first discovered, the Brazilian Babylonians,

131
00:12:17,800 --> 00:12:23,530
no doubt about it, is only the Chinese are going through pretty much the whole of the planet.

132
00:12:24,050 --> 00:12:29,570
All you need is a little insects, which actually just drive away warring mathematicians.

133
00:12:29,570 --> 00:12:36,640
This and this little insect actually discovered that the planes were incredibly effective for its evolutionary survival.

134
00:12:37,360 --> 00:12:43,060
So this is called an extremely strange voice like landed in the forests of North America.

135
00:12:43,750 --> 00:12:54,040
But this particular scholar has a 17 year old's say it on the ground, doing absolutely nothing for 17 years.

136
00:12:54,280 --> 00:12:58,329
And then in the 17th year, all of these cicadas emerge en masse.

137
00:12:58,330 --> 00:13:00,490
And here's the sound of one of these cicadas.

138
00:13:01,120 --> 00:13:08,799
You have to multiply this by several million of these which appear in reports residency levels absolutely

139
00:13:08,800 --> 00:13:16,150
unbearable because they cause you why they so these they they make like eggs and they after six weeks,

140
00:13:16,390 --> 00:13:25,440
six weeks, they all die. For us as Colombian for another 17 years for the next generation of women around us.

141
00:13:25,480 --> 00:13:30,960
Extraordinary lifestyle. First of all, how they managed to count the 17 years gone by.

142
00:13:30,990 --> 00:13:35,309
There's nothing really in the Maxim cycle which has a 17 year period.

143
00:13:35,310 --> 00:13:41,700
So it's sunspots, 11 years, but it's so the fact they even count 17 is pretty impressive.

144
00:13:42,300 --> 00:13:50,580
Very rarely do any appear early or length. But what's curious to me is I 17 that's one of these prime numbers is that importance.

145
00:13:51,210 --> 00:13:58,080
What is seems to be important because there's another species in another area of North America which has a 30 year life cycle.

146
00:13:58,770 --> 00:14:04,920
So you only get these 30 or 70. There's never any 12 year cycle or 14, 15, 16 or 18.

147
00:14:05,370 --> 00:14:09,960
So what is it about these primes, which seems to be in some way helping these cicadas?

148
00:14:11,010 --> 00:14:14,840
Well, we're not really too sure, but we have a theory of this, Mary,

149
00:14:14,850 --> 00:14:21,360
is that maybe there was a phrase around that used to sort of compete to the bee to try and write it

150
00:14:21,360 --> 00:14:26,490
for us at the same time as the Second and the the predator used to appear periodically as well.

151
00:14:26,930 --> 00:14:32,100
And is it called? Is that a plane noise? Sounds like a king out of sync with this predator.

152
00:14:32,250 --> 00:14:35,730
Much better than the second. Is it a non flying license?

153
00:14:36,450 --> 00:14:43,769
Let's take an example. Suppose a predator appears every six years in all forest nesting cicadas.

154
00:14:43,770 --> 00:14:48,150
You've chosen nine years who appear every nine years in the forest.

155
00:14:48,450 --> 00:14:52,889
Well, six and nine. They share a divide.

156
00:14:52,890 --> 00:14:56,150
Very common. So, in fact, you find that it's every second time.

157
00:14:56,150 --> 00:14:59,219
The second is a man pair. Yeah, 18 is a number.

158
00:14:59,220 --> 00:15:07,320
There is a six and nine. And so every 18 is like the cycle of the predator and it very quickly gets wiped out.

159
00:15:07,620 --> 00:15:13,230
But if you change the cicadas lifecycle just a little bit, let's change it to 70 is a prime.

160
00:15:13,650 --> 00:15:20,430
So actually it's appearing more often in the forests and maybe opening itself up to even more attack by the Predator.

161
00:15:20,610 --> 00:15:25,440
But no, because salmon is now prime, it means that six and seven,

162
00:15:25,590 --> 00:15:31,080
those two keep out of sync until it's only a year 42 that they meet for the first time.

163
00:15:31,530 --> 00:15:40,020
So it seems as if the primes gave those had a prime lifecycle has an evolutionary advantage to why you couldn't avoid the predator.

164
00:15:41,160 --> 00:15:47,720
And in the forest in America over the 70 year cycle, it seems like there's quite a competition for who to find the brain.

165
00:15:47,730 --> 00:15:54,780
So maybe the the predators go back in sync found that seven and then as it works harder to find a new prime.

166
00:15:55,620 --> 00:15:58,890
And so I think there's a message here you know your maths you survive in this world.

167
00:15:59,050 --> 00:16:03,480
We have no evidence of this predator, but it's quite a curious life cycle.

168
00:16:03,480 --> 00:16:11,370
So one could say that these cicadas, which still exist in North America, there was a big brood this summer which came out some on the East Coast.

169
00:16:12,030 --> 00:16:18,090
I got a chance to go and see some of these 13 year cicadas a few years ago appearing in the Baltimore region.

170
00:16:18,690 --> 00:16:24,540
They're absolutely extraordinary. And my if I say one insect and the oldest on it, it would be this one.

171
00:16:25,770 --> 00:16:33,239
But, yes, I think if you're going to say which culture really started to understand the problems, and I always think your first answer was that right.

172
00:16:33,240 --> 00:16:37,740
I think the ancient Greeks are really credited with the first explorations.

173
00:16:37,920 --> 00:16:40,739
All the problems and the first real beginning of mathematics,

174
00:16:40,740 --> 00:16:48,480
I think got an analytic subject the ability to improve things with 100% certainty, which is somehow what makes my body so different.

175
00:16:48,630 --> 00:16:56,370
I think to the other sciences and in particular, one of my favourites proves that All the time was about primes by Euclid.

176
00:16:57,090 --> 00:17:05,459
You see the chemists, they have this thing in the periodic table and they have in devised any chemistry classroom or lab and

177
00:17:05,460 --> 00:17:10,470
it's up there on the wall and you've got a list of just all the atoms which are the middle molecules.

178
00:17:10,680 --> 00:17:16,709
So maybe enormous know and I suppose we should have a sort of periodic table prime primes should just write the primes in the table.

179
00:17:16,710 --> 00:17:19,530
And when we need to know something about primes and God can solve it,

180
00:17:20,430 --> 00:17:29,910
but you can show you that's trying to produce a poster with all the primes on voice you've been writing for hour because he proved that the primes,

181
00:17:30,510 --> 00:17:35,880
they never run out. There are infinitely many of these building blocks of mathematics.

182
00:17:36,750 --> 00:17:39,780
But how do you have enough spine on a human?

183
00:17:39,780 --> 00:17:42,810
Beings proved that something will go on to exist.

184
00:17:44,310 --> 00:17:47,639
Well, it was amazing that along the way he came up with and I want to show it to you,

185
00:17:47,640 --> 00:17:52,830
because I think it captures the power of lunch because something is amazing is intimacy.

186
00:17:53,520 --> 00:18:00,630
So his first assumption was, well, supposing we're only following you, perhaps you could write down a list of all the crimes.

187
00:18:00,750 --> 00:18:05,970
Suppose there are there are actually only two of the 43.

188
00:18:06,120 --> 00:18:15,870
Perhaps using those, you could make all those. You know, why didn't we make 5.3 on five times 768 making the different combination?

189
00:18:15,930 --> 00:18:21,870
Perhaps you can make all numbers by not lying easy to you that show you always be missing a number.

190
00:18:22,470 --> 00:18:29,750
How do you do this? We'll take all of those building blocks and multiply them together and then enact a genius us to add ones that numbers.

191
00:18:30,450 --> 00:18:35,490
Okay, so here's a new novel. What primes is that divisible by?

192
00:18:35,790 --> 00:18:42,670
It's got to be. It is why it's either prime or whatever it is. So we said the only primes are in all this, so it must be divisible by one.

193
00:18:42,690 --> 00:18:49,020
That is fine. That is. Which one is it? Well, it's being constructed, so you always get related to one.

194
00:18:49,080 --> 00:18:52,139
This is Euclid's clever trick. So 17, 17.

195
00:18:52,140 --> 00:18:53,940
Divide that. Now you get remainder one.

196
00:18:54,840 --> 00:19:01,290
So it means that most of Mr. Primes this number is either the Prime or Philosophise surprise, which are not on the list.

197
00:19:01,860 --> 00:19:05,290
So you might say, okay, well then why are you asking?

198
00:19:06,030 --> 00:19:10,829
And so you add some more Brian to the list and then you say, Well, that's still number.

199
00:19:10,830 --> 00:19:14,550
You can't build out the prime units because he plays the same trick again.

200
00:19:15,180 --> 00:19:17,240
So I have. However long you try,

201
00:19:17,280 --> 00:19:26,680
Euclid is able to show that you must have missed him on a finite supply and would never be able to build all numbers at this point.

202
00:19:26,700 --> 00:19:30,000
Little bit argument shows the primes as they go on.

203
00:19:30,270 --> 00:19:35,430
So with mathematicians, we have it much harder than the chemists and you see a gradual down on the table.

204
00:19:35,680 --> 00:19:39,330
We've got to find some other way to understand this sequence of numbers.

205
00:19:39,660 --> 00:19:45,239
Now, this thing you might think, oh, maybe you could come up with a clever way to actually find primes,

206
00:19:45,240 --> 00:19:48,580
because you can multiply that together and add one and got a new primes.

207
00:19:48,710 --> 00:19:52,760
Maybe that's a good way of learning. But fortunately, that doesn't always work.

208
00:19:52,890 --> 00:19:54,810
If you do two things three times, fine.

209
00:19:54,930 --> 00:20:06,240
Add one, all that surprising, but very quickly you multiply two times three all the way out to 13, for example, and add one that's multiplying.

210
00:20:06,960 --> 00:20:11,340
It's not busy widening of the prime solve to 13, but it is not prime itself.

211
00:20:11,550 --> 00:20:13,290
And you're right, it is an open question.

212
00:20:13,710 --> 00:20:19,670
You will make your name in mathematical history and you can prove that in the afternoon you get some primes out of this consumption.

213
00:20:19,680 --> 00:20:25,550
We don't know that. So as she's ever since you leave spring, let me have the clients go over forever.

214
00:20:26,050 --> 00:20:30,220
We just don't know where they are. We don't have a formula for how to guarantee you a prime.

215
00:20:30,760 --> 00:20:35,210
Now, having some form is some help in finding some of the biggest crimes we know so far.

216
00:20:35,230 --> 00:20:40,180
So a few years ago, there's something called the Great Internet Mersenne Prime Search,

217
00:20:40,180 --> 00:20:47,110
which you can take possibly you can download a piece of software which you get some computer files on my computer.

218
00:20:47,110 --> 00:20:52,780
Sitting up here on the desk is actually running this program, always in the background, looking for new primes.

219
00:20:53,050 --> 00:21:00,390
And in fact, the person who found this prime, they broke the record having no records, Spergel says.

220
00:21:00,430 --> 00:21:05,499
Could you find a prime with more than 10 million digits, that kind of challenge out there?

221
00:21:05,500 --> 00:21:09,190
And somebody had offered a prize of $100,000, if you can find it.

222
00:21:09,760 --> 00:21:12,770
Now, this is a very good formula for finding big primes.

223
00:21:12,790 --> 00:21:20,380
It was first discovered by a French mathematician, which is why this is called the Great Internet Mersenne Primes.

224
00:21:20,830 --> 00:21:26,380
So you raise two to the power, in this case, 43,112,609.

225
00:21:26,680 --> 00:21:33,009
That's an incredibly reasonable number. But then you take one of that number and suddenly this number then becomes prime.

226
00:21:33,010 --> 00:21:39,810
And it was the first prime to be discovered. You pass this 10 million digit and the person $100,000.

227
00:21:39,820 --> 00:21:44,220
The finding is but is subsequently that was discovered in 2009.

228
00:21:44,230 --> 00:21:45,340
So that's quite a big gap.

229
00:21:45,910 --> 00:21:56,770
Just recently, the beginning of 2013, an even bigger coin was found using the same program and this is now over 17 million digits.

230
00:21:56,890 --> 00:22:02,010
If I read this out, allows you to find it would be here for a couple of months to find out.

231
00:22:04,060 --> 00:22:12,310
But essentially that but thanks to you, we know that there are primes because we want this is a net and that's the best we know so far.

232
00:22:12,320 --> 00:22:19,780
But it's always going to be a big one. So I see there's a prize at $200,000 to the first person to find it a prime with a billion digits,

233
00:22:20,290 --> 00:22:27,790
although I think an interesting will probably use out the whole money if you try to use to your computer an Amazon Prime search to find it.

234
00:22:28,690 --> 00:22:37,299
But I think part of this is just how little we understand about these numbers, because if I asked you to find a billion digits Fibonacci number,

235
00:22:37,300 --> 00:22:38,440
you could read easily,

236
00:22:38,440 --> 00:22:46,749
use the fact that what we know happened in large numbers to find out or a triangular number or a swing of the prime base numbers.

237
00:22:46,750 --> 00:22:51,640
And also we just don't know where they are beyond here, complete mystery.

238
00:22:53,720 --> 00:22:59,030
So this is the challenge, how we find a way to understand how these primes,

239
00:22:59,030 --> 00:23:03,950
the building blocks of the whole of mathematics are laid out through the universe and numbers.

240
00:23:04,760 --> 00:23:10,670
So for centuries, really trying to find formulas or patterns and we just can't seem to get anywhere.

241
00:23:11,900 --> 00:23:18,410
Now our online is quotes, which comes from one of the experts in fine numbers at Princeton and Rico on the air.

242
00:23:18,540 --> 00:23:22,790
And I think that rather sums up one of the traits of being a good mathematician.

243
00:23:23,240 --> 00:23:29,420
He said, When things get too complicated, sometimes my sense is simple wonder how do I ask the right question?

244
00:23:30,230 --> 00:23:35,330
And is that skill to do lateral thinking, which I think makes a great research mathematician.

245
00:23:35,510 --> 00:23:39,620
So we were trying to find the formula to try to find a pattern, but maybe we needed to.

246
00:23:39,650 --> 00:23:45,230
Also, we knew about the primes. I think there's a way to understand what their secrets are.

247
00:23:45,590 --> 00:23:49,460
And it was, in fact, one of the greatest mathematicians, I think in history of all subjects,

248
00:23:49,940 --> 00:23:56,090
Carl Friedrich Gauss, who often knew question about plants and found a hidden pattern.

249
00:23:57,150 --> 00:24:01,580
It's kind of curious. Gauss clearly was going to be a great mathematician from a very young age.

250
00:24:01,580 --> 00:24:10,670
He was a mathematical diary where he recorded his discovery of a useful way to construct a 17 sided figure for his 15th birthday.

251
00:24:11,150 --> 00:24:16,340
He given a book of logarithms, researched that kind of thing you might get from birthdays.

252
00:24:17,540 --> 00:24:22,670
But what was curious was in the back of this book of logarithms was a table of prime numbers.

253
00:24:23,510 --> 00:24:27,860
And Gauss. He understood the rooms, but the prime numbers began to sense.

254
00:24:28,100 --> 00:24:32,270
And he was saying, well, know, what is the secret to the numbers that are in his title?

255
00:24:32,420 --> 00:24:37,730
He couldn't find names, and then he thought it was something different about these numbers.

256
00:24:38,300 --> 00:24:44,000
And he also knew the question is new, this new question which revealed the person rather remarkably,

257
00:24:44,000 --> 00:24:51,260
it revealed a pattern between the price, the faculty school and the logarithms at the front of his book.

258
00:24:52,100 --> 00:24:53,240
So what was the new question?

259
00:24:54,020 --> 00:25:00,560
Well, the new question was, okay, so trying to find where the next time is, let's try to count how many primes there are.

260
00:25:01,250 --> 00:25:07,370
Now, you might say, well, isn't that a bit of a silly thing to do? Because Euclid already proved that there are many, many frauds.

261
00:25:07,370 --> 00:25:14,870
So how can you count them but now say no, I don't know how many primes they're all saying the first ten numbers were two three,

262
00:25:14,870 --> 00:25:18,239
five, 734 primes in the first ten numbers. I see.

263
00:25:18,240 --> 00:25:21,200
My guess is counting one was counted as a prime number.

264
00:25:21,200 --> 00:25:29,210
And that's how you I've seen it in his book of logarithms one in the 19th beginning of the 19th century, with Stewart on it as a prime.

265
00:25:29,390 --> 00:25:32,540
But because we now believe primes for the most important thing,

266
00:25:32,540 --> 00:25:37,130
rob them of their building blocks, they make new numbers, and then you multiply it by one.

267
00:25:37,430 --> 00:25:44,520
You don't get anything new. So it's a little bit like the fact you say so actually we don't say one seven now, so we say Ready for Prime ten.

268
00:25:44,640 --> 00:25:54,340
And that whole piece that I showed you will be a twin 29 times, all the way up to 125 times as a single 6117.

269
00:25:54,350 --> 00:25:58,610
Okay. As in, how far can you predict how many primes you expect to get?

270
00:25:58,850 --> 00:26:02,190
Hundreds of thousand, for example. And you can draw a graph of this.

271
00:26:02,190 --> 00:26:07,759
So sort of a graph which counts the number of points and what I call it counts a staircase in the prime.

272
00:26:07,760 --> 00:26:13,660
So every time you take a step upwards. So, for example, 101 is a prime number.

273
00:26:13,670 --> 00:26:18,170
So you've got 25 rises and 100. When you got to 101, you take a new step up.

274
00:26:18,830 --> 00:26:24,920
So recorded inside this graph is all the information about where the primes up, but now says, okay,

275
00:26:24,950 --> 00:26:30,469
we got obsessed with the minutiae of when you step so this say to step backwards and see if we

276
00:26:30,470 --> 00:26:36,050
can see any overarching patterns the way the staircase grows as they climb higher and higher.

277
00:26:36,620 --> 00:26:41,190
And when he started to look at this question, he's found this constant beginning to match.

278
00:26:41,450 --> 00:26:42,980
And he made this connection with Lawrence.

279
00:26:43,160 --> 00:26:51,440
Now, I just want to check that you all are up to speed with remembering what a rate I mean, how many people used to set of all tangles and errors.

280
00:26:51,460 --> 00:26:55,760
So yes, it will people now putting up their hands.

281
00:26:55,760 --> 00:27:03,320
I should also admits to using a small table because the lower the tables were basically the calculator of their day.

282
00:27:03,860 --> 00:27:09,179
They enable add engineers or navigators to speed up a calculation because this is

283
00:27:09,180 --> 00:27:13,399
the one thing you just have to remember after the logarithm is logarithm times.

284
00:27:13,400 --> 00:27:18,950
Multiplication, intimidation and multiplying two numbers together is quite complicated.

285
00:27:19,160 --> 00:27:23,959
But he did say he did ask for an easy and then can use a little to back so that's

286
00:27:23,960 --> 00:27:28,630
all you have to remember that logarithms to understand this connection the calcite.

287
00:27:29,360 --> 00:27:35,629
So Kells started to analyse the data that he has. This is a lot more data than Gauss had available to him, but they didn't have.

288
00:27:35,630 --> 00:27:41,810
The time was enough to detect this pattern in the number of primes you get as you count higher and higher.

289
00:27:42,770 --> 00:27:50,389
So so let's look at it. So the middle column tells you the number of primes less than the number in the column on the left hand side.

290
00:27:50,390 --> 00:27:54,470
So for example, under a thousand, there are 168 primes.

291
00:27:54,620 --> 00:27:57,830
But the more interesting thing when Posner is in the last column,

292
00:27:58,130 --> 00:28:03,470
because the last quantum record is the proportion of primes amongst all numbers to a thousand.

293
00:28:04,000 --> 00:28:07,520
It's also a thousand with a heart, he prime says.

294
00:28:07,520 --> 00:28:11,330
In a thousand, that's approximately one in six number.

295
00:28:12,280 --> 00:28:20,530
Up to a thousand is at a prime. So we saw from 120 flights are waiting for numbers, up to 100 to prime.

296
00:28:20,740 --> 00:28:24,460
But it's also getting rarer. So up to a thousand is one in six.

297
00:28:24,730 --> 00:28:30,110
And this is one of them all. Scotland is recording. It's recording the proportion of primes to all numbers.

298
00:28:30,730 --> 00:28:34,300
So for example, let's take I had in London.

299
00:28:34,330 --> 00:28:37,510
So I want to talk about numbers about ten, 10 million.

300
00:28:37,520 --> 00:28:42,220
So what was the chance that my London telephone number would be a prime number?

301
00:28:42,610 --> 00:28:50,110
Well, that last one tells you up to 10 million. That's a one in 15 chance that my telephone number is going to be prime.

302
00:28:50,800 --> 00:28:54,210
And I lived a few years ago, and I had to change my phone number.

303
00:28:54,340 --> 00:29:00,220
My telephone number. And I, I phoned up the telephone company and said, I need to be telephone number.

304
00:29:00,490 --> 00:29:06,760
And then she the woman who had got to give a number and I very quickly chatted on my computer and it wasn't prime.

305
00:29:06,880 --> 00:29:14,950
I never remember that. Could I have another one? So she tried another one and I'll pull it out and it is still on prime.

306
00:29:14,950 --> 00:29:20,020
And we went through this column trying to make sure she just got herself images and then thought,

307
00:29:20,140 --> 00:29:25,020
I'm going to be the next door which comes out and it's almost going to have an even number.

308
00:29:26,620 --> 00:29:32,680
But this tells me we would have to go through this 50 times and one of those might be the product.

309
00:29:32,710 --> 00:29:34,400
So I'm still here.

310
00:29:35,570 --> 00:29:45,070
So the last column is where the pattern that Gauss discovers emerges, because all the while the Prime Minister isn't there watching the beginning.

311
00:29:45,070 --> 00:29:50,380
So it's all the same as the sequence we had. Sometimes things need a bit of time to settle down.

312
00:29:50,590 --> 00:29:58,930
This one is a little sound around ten 10,000. So there's a one in 8.1 chance that a number is prime around 10,000.

313
00:29:59,500 --> 00:30:03,550
And when you multiply the first one by ten go 10000 to 100000.

314
00:30:03,950 --> 00:30:08,649
The chance of getting a decline goes down. So it's now one in 10.49.

315
00:30:08,650 --> 00:30:16,330
It's 2.3 to the chances. If I what I want you to think about is to think about this a little bit like that in nature.

316
00:30:16,990 --> 00:30:19,960
Use some dice to actually choose the problems.

317
00:30:20,380 --> 00:30:29,530
And this last column records the number of signs on the dice that you need when you're trying to choose the crimes around a thousand.

318
00:30:29,530 --> 00:30:37,320
So why don't you say so? So you could use a cube. A six is faced shape with one side of the opinion.

319
00:30:37,660 --> 00:30:44,720
And to get a sequence of numbers, which looks very much like primes around a thousand, you could use this and just circle numbers every time.

320
00:30:44,760 --> 00:30:50,110
Get a P and you get a sequence of numbers to very much like with primes around a thousand,

321
00:30:50,380 --> 00:30:54,310
but as in plain white, the number of signs on the dice increases.

322
00:30:54,320 --> 00:30:59,799
So we see that there are around 10,008.1 and that's a bit like having an octahedron.

323
00:30:59,800 --> 00:31:03,040
So this is an eight face figure. Just one side of the prime.

324
00:31:03,040 --> 00:31:06,970
And this is the one that might be deciding primes around 10,000.

325
00:31:07,300 --> 00:31:13,240
And as you go further up, so on here, around a million, you've got a 12.7.

326
00:31:13,240 --> 00:31:16,390
So there's like 12 sided dice. So don't count Hedren.

327
00:31:16,570 --> 00:31:19,990
This might be choosing the crimes around 10,000.

328
00:31:19,990 --> 00:31:27,910
So so the trick is, well, how can you predict the number of sides in the dice as you climb higher and lower angles?

329
00:31:27,910 --> 00:31:35,920
Notice this concept every time you multiply the first column by ten, the number of signs in the dice seems to be going up by 2.3 each time.

330
00:31:36,160 --> 00:31:43,720
8.1 to 10.4 you add 2.3 multiply by ten again 10.4 to 12.7 adding 2.3.

331
00:31:44,350 --> 00:31:48,669
Now you get 15.0 to 17.4. That's just a random rounding up.

332
00:31:48,670 --> 00:31:55,180
This happens and essentially this pattern just keeps on going the way the primes do now,

333
00:31:55,180 --> 00:32:00,770
although on a small scale they look very random when you take a step back because

334
00:32:00,790 --> 00:32:06,949
they seem to have this incredible pattern being just sort of chosen by dice numbers.

335
00:32:06,950 --> 00:32:10,900
Size is decided by adding 2.3 every time you multiply by ten.

336
00:32:11,650 --> 00:32:14,830
But here's the connection with the lower because remember logarithms turn

337
00:32:14,830 --> 00:32:21,760
multiplication into addition by multiplying the best column by ten I add 2.3.

338
00:32:21,760 --> 00:32:29,440
So the number size of the dice is the lower rhythm function, which is controlling the number of signs in the dice as you climb higher and higher.

339
00:32:29,530 --> 00:32:37,350
That was Gauss's belief. So you want to know the probability that the number is prime that is is lower than function, which doesn't.

340
00:32:37,960 --> 00:32:42,190
So in some sense logarithms tell you the numbers on the dice.

341
00:32:42,220 --> 00:32:50,050
So if I want to use a very large number, the probability that is prime is one in logarithm of that number.

342
00:32:50,380 --> 00:32:57,760
I don't believe that this wasn't you know, we got to be careful because as I shown, this sequence can have a pattern.

343
00:32:57,760 --> 00:33:03,250
It looks like powers of two when I was doing these numbers for you and it suddenly went to 31 and 32.

344
00:33:03,700 --> 00:33:07,960
So you have to be careful. But 100 years off, the Gauss made this prediction.

345
00:33:08,440 --> 00:33:11,260
It was in fact confirmed that this is the right model for the.

346
00:33:11,360 --> 00:33:16,450
Trying set out in this incredibly useful way, however far you go up in the universe, anomalous.

347
00:33:16,790 --> 00:33:22,190
And in fact, you can use this model now to make a prediction about the number of crimes that you'll see.

348
00:33:22,370 --> 00:33:30,770
Because, for example, if my friend is three times out of six, you find that this says things being prime numbers.

349
00:33:31,070 --> 00:33:37,660
So I can use the fact that I know how many sides there are on the dice to predict how many crimes I should get because I call it paranoia.

350
00:33:38,270 --> 00:33:41,900
And so you can use this gauss. This is amazing. I just.

351
00:33:46,410 --> 00:33:50,250
I think it's rather amazing that a 15 year old boy, Gauss,

352
00:33:50,250 --> 00:33:58,470
discovered this extraordinary pattern that seems to be hiding behind waypoints in our lifeline slightly refines this approximation.

353
00:33:58,710 --> 00:34:04,530
And so what he got was the overarching trend of the statements of the problem as you climb higher and higher.

354
00:34:05,310 --> 00:34:09,550
Now, this is pretty good. Good enough for an engineer, but not good enough for a mathematician.

355
00:34:09,570 --> 00:34:17,129
We don't have things to be precise. So how can we get from gauss's guess over arching sort of positing that line?

356
00:34:17,130 --> 00:34:23,670
So an exact formula to the number of primes because actually from 1000 to 1030, there aren't five primes.

357
00:34:23,670 --> 00:34:27,660
There are actually I think only four or six is slightly different.

358
00:34:28,140 --> 00:34:36,330
So you always think it's an error. And of course, if you throw it, you're not going to expect to get exactly a six times landing on the wrong side.

359
00:34:36,660 --> 00:34:42,300
So how do we work out the errors in these times? We've got a good sense of the pattern, but how do we correct this?

360
00:34:43,970 --> 00:34:44,450
Well, it wasn't.

361
00:34:44,450 --> 00:34:53,780
Gauss began as a student, then Harem, who came up with an extraordinary discovery about how to correct the errors that Gauss assessment will make.

362
00:34:54,350 --> 00:35:02,450
And the best way, I think, to explain Reading's ideas, which are full of extraordinary mathematics involving complex numbers,

363
00:35:02,450 --> 00:35:05,150
imaginary numbers, I would say function Z, the functions.

364
00:35:05,390 --> 00:35:12,560
But actually I think the way you get to the heart of what Raymond did and to understand the what we now call the Riemann hypothesis,

365
00:35:12,560 --> 00:35:17,030
the biggest problem in the whole of mathematics is the music,

366
00:35:17,150 --> 00:35:22,430
because actually what Freeman discovered was the hidden music underlying the finds,

367
00:35:22,430 --> 00:35:27,380
which helps you to understand ebb and flow of these dances and choosing the primes.

368
00:35:27,650 --> 00:35:37,570
So what I do. It's. I could explain to you a little bit about how musical theory works.

369
00:35:37,810 --> 00:35:42,030
We understand it's a lot of science. The ideas that we've developed.

370
00:35:42,530 --> 00:35:51,220
So actually the building of some numbers, applying the building blocks of music discovered by for a reasonably similar period.

371
00:35:51,850 --> 00:35:55,259
This is how the tuning fork makes. So there's a sound.

372
00:35:55,260 --> 00:36:06,810
But, you know, I've got a tuning fork here. So that is actually the purest sort of sound because if you are cool, that sound on a similar scale,

373
00:36:07,080 --> 00:36:15,420
what you get is a perfect sine wave and well for a start it is a sideways eruption, the building blocks of all sounds.

374
00:36:15,960 --> 00:36:16,890
So for example,

375
00:36:16,950 --> 00:36:27,540
if I take a different instrument so upon in so there's the eye on the ball in I'm playing an I on hear this is I have to borrow only my twin violin.

376
00:36:27,540 --> 00:36:32,770
So this is why this is the less half size. So it's the same.

377
00:36:36,190 --> 00:36:42,640
But it's very difficult to use them. So what's going on here? I mean, if I were to reproduce, I see my computer doesn't have all these.

378
00:36:43,030 --> 00:36:47,440
Well, how does the computer manage to make the sound of a body come out of the speakers?

379
00:36:48,160 --> 00:36:55,450
Well, what we understood is the volume is made up not of just a single tuning for a that you heard earlier, but many of you.

380
00:36:56,290 --> 00:37:00,130
So there are many sideways that build up to the sound of evolving.

381
00:37:00,280 --> 00:37:09,730
So I think if you recall the sound of on and on a citizen, you get this kind of jagged sawtooth growth, but you can build this straw out of sideways.

382
00:37:09,880 --> 00:37:15,050
So essentially the same waves that you get are all the sound waves that fit the length of the morning.

383
00:37:15,070 --> 00:37:21,940
So the annual hearing is actually the very sort of big sine wave that's like racing between the beach and the the body.

384
00:37:22,240 --> 00:37:28,319
But we can also fix another sine wave inside that which is vibrating twice as fast and as half the wavelength.

385
00:37:28,320 --> 00:37:36,400
So it's sort of vibrating in these two places. So that's called the first call warning, and that's also the period that notes the same time.

386
00:37:36,670 --> 00:37:40,600
You're also hearing any sine wave that can fit inside the length of this violin.

387
00:37:40,870 --> 00:37:44,290
So, in fact, you're hearing ones that have half the wavelength of the wavelength.

388
00:37:44,470 --> 00:37:50,470
And it's the culmination of all these tuning forks which combined make you hear the sound move on it.

389
00:37:50,500 --> 00:37:54,790
So we can do this radically by combining. So here's the sine wave.

390
00:37:54,910 --> 00:38:01,240
This is the the, the a of the tuning. But now I'm going to add on some other notes as well.

391
00:38:01,270 --> 00:38:05,150
I'm going to add on the first time. So that's a sign letter which is going to vibrate.

392
00:38:05,620 --> 00:38:09,940
So it's going to what it's going to do is push and see. Yes.

393
00:38:10,120 --> 00:38:15,970
So if I add on another sign which vibrates twice as fast as going up there and then down here.

394
00:38:16,480 --> 00:38:20,260
So what's going to happen is that when I add the graph, it's going to push this little graph up,

395
00:38:20,830 --> 00:38:25,690
but underneath here is negative and it's going to pull this side of the graph down.

396
00:38:25,720 --> 00:38:30,760
So what I'm going to do is add all of these lines and the first one, I'm going to add more and more sideways.

397
00:38:30,940 --> 00:38:35,740
And as we add on more of this, so I might add more of this line.

398
00:38:36,190 --> 00:38:44,650
Let's run this little animation. You see that the sound of the tuning pool gradually changes in the sawtooth shape of the body.

399
00:38:45,070 --> 00:38:48,620
And this is really what the speakers are doing when they reproduce the sound of a violin.

400
00:38:48,910 --> 00:38:52,470
They're just raising it all of these different sine wave scanner tricks here.

401
00:38:52,480 --> 00:39:00,459
And hearing this shape, which is the shape of the body that I'm a trumpeter rather than a violinist.

402
00:39:00,460 --> 00:39:05,350
And harmonics are these things like absolutely essential to the way a trumpet works.

403
00:39:06,610 --> 00:39:11,970
Because as trumpet so an early now you didn't have any vowels at all.

404
00:39:11,980 --> 00:39:15,970
You couldn't change the length of the piece of string, as it were, the length of the tune.

405
00:39:16,690 --> 00:39:23,290
You had to create new notes by just putting more energy through the the point, and that will create the higher harmonics.

406
00:39:23,290 --> 00:39:27,070
So, for example, I don't have to change the length of the point, but I can get many notes out of this.

407
00:39:27,250 --> 00:39:31,210
What I'm doing is getting the different sideways, which fits inside the trumpet.

408
00:39:33,040 --> 00:39:46,790
But. Uh, so that's the sound, although essentially all those different sine waves that can fit in that length of point and so thrashy.

409
00:39:47,190 --> 00:39:50,620
I mean, there are only eight different combinations of the vowels here.

410
00:39:50,630 --> 00:39:54,410
I mean, you see some harmonics every time I play the trumpet.

411
00:39:55,040 --> 00:40:00,980
So I don't think you'd be surprised if you have a switch that's gonna.

412
00:40:03,470 --> 00:40:35,900
One, two, three, four, five. I prefer.

413
00:40:41,200 --> 00:40:55,629
I. So I had a bit of time to warm up, but I didn't change the instruments.

414
00:40:55,630 --> 00:41:00,010
So you're going to need different combinations of sideways to create the sound of that instrument.

415
00:41:00,020 --> 00:41:07,900
So, for example, I'm going to take the clarinets. So the sound of the clarinet is used here while I travel with Chinese.

416
00:41:12,520 --> 00:41:15,579
It's impossible to close down, as you can see now.

417
00:41:15,580 --> 00:41:19,540
And you see you got these kind of square like coronations of a castle.

418
00:41:19,810 --> 00:41:23,940
And this is because you're getting different sorts of sideways signs here.

419
00:41:24,490 --> 00:41:31,030
So again, you got basically the shooting pool, which is so that the sun is going to be open at one end and closed in the other.

420
00:41:31,030 --> 00:41:37,860
So you get sort of whole the sine wave and then the next sine wave that appears inside here is actually one which is vibrates three times as follows.

421
00:41:37,880 --> 00:41:43,630
So you get this, this shape. And so you're not hearing all this the same sine wave of all that is fine,

422
00:41:43,810 --> 00:41:48,300
but about half of them actually only there is this sort of odd number, wavelets.

423
00:41:48,460 --> 00:41:52,000
And if you combine these now, you've got a very different shape graph.

424
00:41:52,000 --> 00:41:58,470
So basically start with the tuning for sine wave and all of these extra functions.

425
00:41:59,050 --> 00:42:04,350
In this case, the same way this library did three times. So it goes up, then down, then up again.

426
00:42:04,360 --> 00:42:07,810
So that's going to pull the middle of the graph down, the science up.

427
00:42:08,050 --> 00:42:12,490
So when I run is an add on the harmonics, I get a different shape going.

428
00:42:12,850 --> 00:42:18,700
So the different combination of sine wave produces a square function and this is the sound of the parent.

429
00:42:18,970 --> 00:42:26,860
And in fact, every time I see if I was basically just taking advantage of the sound of the orchestra being broken down into the sideways,

430
00:42:26,860 --> 00:42:30,310
the different frequencies to make up the the graph of that sound.

431
00:42:32,050 --> 00:42:36,220
So this is the idea or how this going to sound because in a way,

432
00:42:36,220 --> 00:42:42,310
what you should think of as the the status of the problems is that it's the music we're trying to capture that down to.

433
00:42:42,310 --> 00:42:46,810
The girl's got a good approximation. He's got this graph which is kind of like that.

434
00:42:46,810 --> 00:42:51,070
Well, that's a bit like the sine wave or the tuning fork is the first approximation.

435
00:42:51,490 --> 00:42:55,780
Riemann's were using the z function and complex numbers.

436
00:42:55,960 --> 00:43:04,000
He managed to find some rather weird waves harmonics hiding behind which you add these harmonics along the different frequencies.

437
00:43:04,150 --> 00:43:09,220
They help you to correct galaxies, get gas and get a true formula for the number of points.

438
00:43:09,670 --> 00:43:13,720
So these are what are called the series of the Riemann Z definition.

439
00:43:13,870 --> 00:43:20,419
But essentially what they give you a graphs which are waves and rather sits like sideways and a little more complicated.

440
00:43:20,420 --> 00:43:26,380
And so the first harmonic is at the third, they're actually many of these on this is the 29th and the 50th.

441
00:43:26,620 --> 00:43:30,610
But inside here, the waves are kind of encoded.

442
00:43:30,870 --> 00:43:34,720
The errors the palace is making when he's making his guess, the number of points.

443
00:43:34,960 --> 00:43:42,760
So I have to add these on Gauss's case to get a formula for the number of points you see here.

444
00:43:47,020 --> 00:43:55,360
What I want to show you now is reminds me of the number of times I set this course out, identifying and identifying choices.

445
00:43:55,840 --> 00:44:02,450
It's actually a cicada. It was called out in the desert island, and I had to take one mathematical formula with me.

446
00:44:02,470 --> 00:44:06,150
I think it will be this looming agreement because I think it is absolutely magical.

447
00:44:06,190 --> 00:44:10,690
So I'm going to say this graphical form or other similar to the the graphs, the music.

448
00:44:10,870 --> 00:44:15,819
So the idea is this we're trying to get a formula for the number of primes.

449
00:44:15,820 --> 00:44:19,510
That's the blue states gauss's case.

450
00:44:19,510 --> 00:44:22,930
Using the logarithm is a good estimate, but it's not exact.

451
00:44:23,500 --> 00:44:30,610
If I want these waves of regions which can sometimes add a little bit all or sometimes type in the way encoded inside,

452
00:44:30,610 --> 00:44:33,650
there are the errors that Gauss is making.

453
00:44:33,670 --> 00:44:39,220
So what I'm going to do is run for you this little animation. So galaxies, gas is a yellow graph.

454
00:44:39,610 --> 00:44:45,759
The blues that I use is the graph we want to capture by adding all these waves which push and pull the graph wrong.

455
00:44:45,760 --> 00:44:52,480
In the same way as the clarinets emerge from the tuning fork on a violin version that you bought by adding the wrong frequencies.

456
00:44:52,720 --> 00:45:03,490
If you add all these types of regions, it gradually pushes and pulls Gauss's case until even over hundreds of these harmonics,

457
00:45:03,490 --> 00:45:08,950
these little waves pushing and pulling the graph, you see the staircase beginning to emerge.

458
00:45:08,950 --> 00:45:17,350
The subtlety of where the primes really are is encoded by these waves that we discover these zeros of the Riemann ze function.

459
00:45:17,590 --> 00:45:25,240
So already, by a hundred ways you're able to tell, oh, there are no primes between 23 and 29, which you couldn't tell with Gauss's guess.

460
00:45:25,900 --> 00:45:29,350
So encoded inside here is really how the pines are laid out.

461
00:45:30,380 --> 00:45:36,590
And it tells us something rather amazing because although the planes outwardly look like they have no pattern at all,

462
00:45:36,680 --> 00:45:42,980
we saw some with his music when remote analyses. He found an amazing pattern hiding behind music.

463
00:45:43,370 --> 00:45:53,029
Because if you think about the same ways, what's important for a CD player is, first of all, the frequency of those waves, but also the amplitude.

464
00:45:53,030 --> 00:45:58,160
How do all that frequencies contribute? The amplitude is how big the wave is.

465
00:45:58,430 --> 00:46:04,819
And Sarina, he quoted a graph which recorded for each of these waves.

466
00:46:04,820 --> 00:46:14,720
What frequency? I want to ask you have the labels so you pulse the frequency will be on the vertical and horizontal will be allowed.

467
00:46:15,860 --> 00:46:23,799
The amazing discovery he made was this The frequencies all changing the volume of these waves is playing.

468
00:46:23,800 --> 00:46:31,340
That seems to be exactly the same. And so this is all of these that seem to be lying on the magic straight line.

469
00:46:31,520 --> 00:46:36,010
They're going to be scattered all over the place. But the music seems to have this amazing pattern inside.

470
00:46:36,510 --> 00:46:39,020
So they lie on this thing, which is will read his critical line.

471
00:46:39,350 --> 00:46:45,230
I really believe it's going to just be a coincidence that first hand all have his just perfect balance too.

472
00:46:45,440 --> 00:46:53,780
And he made this conjecture. He believed that all of the notes that he discovered they would all be playing in this perfect kind of balance.

473
00:46:53,990 --> 00:47:00,980
And this is what we call the Riemann hypothesis. We are not being able to prove it is one of the biggest open problems in the whole of our subject.

474
00:47:01,580 --> 00:47:09,920
And it's really this incredible pattern in the music actually explains outwardly why the primes do look so random.

475
00:47:10,820 --> 00:47:14,950
If one of these notes was off this line, it would be it.

476
00:47:14,980 --> 00:47:20,390
Since playing much louder than all of the other notes. Read the material slightly, very nicely, he said.

477
00:47:20,600 --> 00:47:26,820
It'd be like listening to an orchestra and then suddenly the tuba comes in and blasts the rest of the orchestra out.

478
00:47:26,840 --> 00:47:30,950
So all you can hear is the tuba. So it's really like all this is is false.

479
00:47:31,160 --> 00:47:36,420
And there's a note off the line. It would imply an incredible presence on the client.

480
00:47:36,450 --> 00:47:43,070
So it's one of those notes controlling how the primes are distributed in a way you can interpret this.

481
00:47:43,370 --> 00:47:47,399
So I think this is rather this is called the Riemann hypothesis.

482
00:47:47,400 --> 00:47:55,640
So it says that all of these notes are playing at the same volume, and the Riemann hypothesis sort of explains why you don't see any presence.

483
00:47:55,790 --> 00:48:06,470
In fact, it will predict for these prime number dice point on all these offensive to all a distributing the primes fairly amongst all of the numbers.

484
00:48:06,710 --> 00:48:09,740
The Riemann hypothesis actually is capturing exactly that.

485
00:48:09,980 --> 00:48:15,530
But the error controlled the error is how big this wave is.

486
00:48:15,800 --> 00:48:21,830
The error that remains predicting is exactly what you would expect from a fair set of dice.

487
00:48:22,220 --> 00:48:26,900
So reminisce hypothesis is capturing in some sense the passage of the primes.

488
00:48:26,900 --> 00:48:32,420
Although they are random, they are as fairly distributed through the universe of numbers as we could hope for.

489
00:48:38,050 --> 00:48:48,280
I often quite like to describe the Riemann hypothesis as almost describing the atoms of molecules in the molecules of air.

490
00:48:49,160 --> 00:48:52,420
Well, we don't know exactly precisely where every single molecule is,

491
00:48:53,290 --> 00:48:58,540
but I do know that if I go to one corner of this room, I'm not going to suddenly find a vacuum and I'm going to die.

492
00:48:58,570 --> 00:49:03,790
I know that the molecules are fairly distributed through this route and there's a chance that

493
00:49:03,790 --> 00:49:08,800
they might be able to use their two parts the way that we think the gas is distributed.

494
00:49:08,950 --> 00:49:14,050
It's not important where each one is. I don't know that there's they're pretty evenly distributed.

495
00:49:14,410 --> 00:49:19,330
And for the primes very often as a mathematician, I'm going to need to know where exactly each point is.

496
00:49:19,540 --> 00:49:24,220
What I need to know is that fairly soon it says I'm climbing up through the universe in numbers.

497
00:49:24,370 --> 00:49:27,430
I'm not going to suddenly find a vacuum without any problems at all.

498
00:49:27,670 --> 00:49:34,540
And that's what the Riemann hypothesis very often helps us. Well, let us kind of ask the question where all of the primes and it tells us enough

499
00:49:34,540 --> 00:49:40,030
about how the primes are distributed to make progress in understanding numbers.

500
00:49:40,370 --> 00:49:46,930
And I think it's an amazing because in a way, you know, nature gave in these numbers the primes and outwardly seem to have no patterns at all.

501
00:49:47,080 --> 00:49:52,150
But the primes are the building blocks of the subject looking for patterns.

502
00:49:52,420 --> 00:49:56,170
But mathematicians, we found some clever ways to look at these primes.

503
00:49:56,350 --> 00:50:02,710
A new way to look through Google gal. They changed into the the logarithm controlling the number of signs in the dice.

504
00:50:02,890 --> 00:50:08,290
Then through the music to be able to discover this kind of hidden music where suddenly this very strong pattern,

505
00:50:08,290 --> 00:50:12,550
the reason believes, is there controls the way that points outwardly look.

506
00:50:12,880 --> 00:50:18,160
So somehow the power of the mathematician is finally finding some order inside these numbers.

507
00:50:18,220 --> 00:50:22,810
Although we still looking for an explanation about why those notes sound like they did.

508
00:50:23,290 --> 00:50:31,570
Is it $1,000,000 prize offered by the Clay Institute for anybody who can prove the primes are really behaving as we even believes they do?

509
00:50:32,560 --> 00:50:37,840
Now I discovered that I study the primes because I think that useful.

510
00:50:37,840 --> 00:50:41,440
It's somehow trying to get to the eternal truths of the universe.

511
00:50:41,890 --> 00:50:46,610
But there are also now very practical reasons for understanding the primes in France.

512
00:50:46,870 --> 00:50:50,020
One of the challenges I gave you is getting these 105 primes.

513
00:50:50,170 --> 00:50:53,230
You managed to pull that together in a business. Three, five, five, nine, seven.

514
00:50:53,380 --> 00:51:02,830
Why isn't that a challenge for Young? And I say offer a price and offer a bottle of champagne to anyone who can crack these numbers on the Leonardo,

515
00:51:02,830 --> 00:51:11,140
which is not prime number 100 with 6000. Okay. Well, I'll give you this one right at 99 one 1911.

516
00:51:12,220 --> 00:51:15,820
I have a feeling that I actually. Okay. So that's a challenge.

517
00:51:16,130 --> 00:51:19,480
Okay. 9,999,911.

518
00:51:19,720 --> 00:51:23,710
You have to find the primes which builds that normal prime number.

519
00:51:24,340 --> 00:51:27,610
So it is made out of sodium chloride and you have to pull it apart.

520
00:51:27,910 --> 00:51:32,200
And so I did offer a bottle of champagne was in the fridge, but it's beginning to warm out the lights.

521
00:51:33,130 --> 00:51:37,660
So the first person to discover two primes to build the number will win the ball.

522
00:51:38,470 --> 00:51:39,730
Now, why is this important?

523
00:51:40,000 --> 00:51:46,990
Well, actually, this problem is now the heart of all the codes that are being used on the Internet to protect your credit cards.

524
00:51:47,140 --> 00:51:55,450
So essentially, if you send your credit cards to a company online and you want to keep that credit card secure, they send you an on the line.

525
00:51:55,450 --> 00:52:03,670
It's actually a little bigger than this. Your computer on a calculation with this number and your credit card number says it all across the wires.

526
00:52:03,760 --> 00:52:10,450
But to do that calculation, you need two primes, which below is number to be able to undo the calculation.

527
00:52:10,690 --> 00:52:18,340
So criminals, this is public on every poll, on any Internet websites is selling you something that would be a public number.

528
00:52:18,520 --> 00:52:24,520
If you can write that public number into the two primes, you'll be able to break every credit card that's going to that website.

529
00:52:25,480 --> 00:52:28,690
So the number is slightly bigger than nine.

530
00:52:28,690 --> 00:52:32,919
And just on to his is the kind of size that we cannot crack at the moment.

531
00:52:32,920 --> 00:52:37,120
So it is a $200,000 prize for anyone who could crack this number.

532
00:52:37,120 --> 00:52:40,570
So if you don't like champagne with money, then this one.

533
00:52:42,250 --> 00:52:50,890
But essentially, you know, the whole of Internet cryptography is based on the fact that we don't really understand these numbers at all.

534
00:52:51,430 --> 00:53:00,490
Chemists, they have a spectrometer which divides chemicals into the constituents, and we do not have a prime number spectrometer in mathematics.

535
00:53:00,890 --> 00:53:04,840
So for me, I think since, you know, I started with question what is mathematics?

536
00:53:05,080 --> 00:53:07,989
And for me, think what makes mathematics a living,

537
00:53:07,990 --> 00:53:16,930
breathing substance is actually these unsolvable Riemann hypothesis and the challenge of trying to find some pattern in the primes.

538
00:53:35,360 --> 00:53:43,040
So we had a chance for some questions. We have some microphones for you, and when you get to my phone, you can stand up for the cameras.

539
00:53:43,550 --> 00:53:50,810
So this is way where you're playing your parts in this television production because we are probably,

540
00:53:50,810 --> 00:53:54,820
since the ensemble is be giving incredibly intelligent questions.

541
00:53:54,830 --> 00:53:59,000
So please help me to live up to that billing.

542
00:53:59,000 --> 00:54:03,530
And any questions from you about problems in mathematics?

543
00:54:03,830 --> 00:54:13,520
You put your hand up now get him on TV. The first one is always the most nerve wracking one in any society we have to dismantle to stand up.

544
00:54:19,070 --> 00:54:22,700
How do you verify that very large price to ensure that they are prime?

545
00:54:24,020 --> 00:54:33,270
Yeah, that's a very good question. There we've chosen these very large primes because they have a very particular property, the Mersenne primes.

546
00:54:33,290 --> 00:54:38,660
So let me solve the Mersenne primes. How do you verify that they are actually prime numbers?

547
00:54:38,990 --> 00:54:42,860
They're a very special sort of number because they have to surpass something at minus one.

548
00:54:43,130 --> 00:54:48,950
And we discovered a way to test their finality, which is very different from just a random number.

549
00:54:49,280 --> 00:54:52,480
In fact, it's something called the Lucas Lima test.

550
00:54:52,490 --> 00:54:58,100
And it's what actually your program will be running on as it's running on your laptop.

551
00:54:58,520 --> 00:55:01,550
And essentially, it's sort of test something much simpler.

552
00:55:02,060 --> 00:55:03,920
It turns the formality.

553
00:55:03,960 --> 00:55:11,820
This is about whether a certain number divides another number in a sequence, which is rather like the Fibonacci sequence and the Lucas Lane numbers.

554
00:55:11,840 --> 00:55:16,700
You have to build these. So it's a very constructive thing which which proves whether that number is prime.

555
00:55:17,270 --> 00:55:22,790
We already know. And it's something that the ancient Greeks knew that if you go to the power of something,

556
00:55:22,790 --> 00:55:27,700
that thing must be prime in order to the power of it minus long have any chance of.

557
00:55:29,060 --> 00:55:34,070
But what about that sort of primes? I mean, for example, every time an alter, if you will,

558
00:55:34,070 --> 00:55:44,660
have in your warnings small columns which are actually probably generating primes when they're being used in order to find to create cryptography.

559
00:55:45,450 --> 00:55:50,940
And so how do we actually find primes of, say, 100 digits, which is sort of a couple of hundred pages?

560
00:55:50,990 --> 00:55:54,410
What do you need to create some these primes that use on the Internet?

561
00:55:55,130 --> 00:56:02,360
And then often it's just a probabilistic we have a way of testing the probability that a number is prime,

562
00:56:02,840 --> 00:56:07,580
which is generally good enough, and it involves something called modular arithmetic.

563
00:56:08,090 --> 00:56:16,430
So we know that primes have a very particular property or clock arithmetic, and it's something that was discovered by Thoma and Euler.

564
00:56:16,940 --> 00:56:24,080
That's and that's what's being used to test whether a number is has a good chance of being prime.

565
00:56:25,940 --> 00:56:32,540
Because one of the problems is that you have to really take one step to find all the numbers up to the square root of the number.

566
00:56:32,540 --> 00:56:39,590
And that's why you're trying to do that with the number I chose, which adds a prize of $200,000.

567
00:56:39,950 --> 00:56:46,760
Just you wouldn't be able to find a prime position. That particular number you're trying to prove that isn't prime by finding primes to builds it.

568
00:56:47,630 --> 00:56:56,860
And so there are some attacks out there which involve quantum computers and we do a lot of quantum computing here in all cases.

569
00:56:57,670 --> 00:57:04,520
Instead of testing one line after another, a quantum Q computer that could potentially test all primes in one go.

570
00:57:04,520 --> 00:57:11,179
Because what happens in quantum physics when you take advantage of it's an electron wrong being just here or here,

571
00:57:11,180 --> 00:57:14,180
on or off can be in two places at the same time.

572
00:57:14,450 --> 00:57:20,629
And you can use that essentially in a quantum computer to to put the quantum computer into a state subsidy test,

573
00:57:20,630 --> 00:57:26,810
all primes in one go and collapses into the states where the primes are the ones that divide that number.

574
00:57:26,810 --> 00:57:31,700
So we have a software that has a hardware. Yes, it creates a quantum computer.

575
00:57:32,000 --> 00:57:37,050
Good enough. I think we can crack 15 into three times five. But I think you from the last.

576
00:57:39,060 --> 00:57:42,520
It's not the best question down here. And yet we've got the mind to.

577
00:57:49,850 --> 00:57:56,450
Send us your beer formula that you will be able to find all the prime numbers.

578
00:57:56,460 --> 00:58:00,880
So obviously a really complicated formula, but you can either be wrong.

579
00:58:02,120 --> 00:58:06,420
That's a good question because it sort of raises the question about what is similar to the formula.

580
00:58:06,440 --> 00:58:14,419
Do you want what are you looking for so you can already prove that that works the same polynomial for me I have.

581
00:58:14,420 --> 00:58:18,889
That's a nice polynomial formula which told you the number of simple division office it was.

582
00:58:18,890 --> 00:58:27,530
You raise numbers to the power four and do it more know, but we can already prove that you can't have a polynomial which will give you all the time.

583
00:58:27,530 --> 00:58:30,830
You always have to be some numbers which it produces. Which. All right.

584
00:58:31,190 --> 00:58:37,610
Well then why not? Found an amazing formula which produce primes much more frequently than you might expect.

585
00:58:37,610 --> 00:58:41,300
We sent out A, B and squared plus and 41.

586
00:58:42,260 --> 00:58:48,580
You run that and try on this and in that you get amazing number of primes coming out of that format.

587
00:58:48,890 --> 00:58:51,200
But you might say, okay, well maybe it is a different format.

588
00:58:51,500 --> 00:58:58,490
And I'm saying, you know, Riemann's formula is does give you a way of understanding the primes,

589
00:58:58,910 --> 00:59:03,920
but it's not a formula that can be used, for example, to find a prime with a billion digits.

590
00:59:03,920 --> 00:59:11,330
But it's a formula that can help you. And so that's a little bit like trying to sing the highest note.

591
00:59:11,960 --> 00:59:19,460
Well, that's what music is. So so the search for the big primes, I think is a little bit serve, and that's not really what mathematics is about.

592
00:59:19,670 --> 00:59:27,050
Much more interesting to me is to try and find what's the overarching way all the pines are laid out in the universe and and read this formula.

593
00:59:27,060 --> 00:59:32,030
Although now to tell you the biggest the prime has been in digits.

594
00:59:32,210 --> 00:59:37,190
It tells us, I think, a lot more. And it tells us about the underlying patterns in the primes.

595
00:59:37,190 --> 00:59:42,320
All that is on another high note. You know, as a Christian, every.

596
00:59:43,460 --> 00:59:48,410
All right. What progress, if any, has been made towards improving the Riemann hypothesis?

597
00:59:49,630 --> 01:00:00,400
Well, there was some exciting progress made about a couple of decades ago when there was an amazing connection with physics.

598
01:00:01,240 --> 01:00:07,420
Because one question you can ask about the Riemann hypothesis is about amplitude of those waves.

599
01:00:08,080 --> 01:00:11,840
But the other thing I quoted was the frequency. So you might ask,

600
01:00:12,220 --> 01:00:20,440
is there any sort of logic to the frequencies that you find of the notes that we have that tell you how the problems are distributed?

601
01:00:21,010 --> 01:00:27,190
And somebody made a statistical analysis of the sort of where these harmonics are happening.

602
01:00:27,610 --> 01:00:33,790
And what they discovered is that you have a random property at the Institute for the Study in Princeton.

603
01:00:34,120 --> 01:00:39,090
So now one of the important things in science is bringing people together from many different disciplines.

604
01:00:39,100 --> 01:00:45,160
This is what we try to do here in Oxford, and that physicists talk to this mathematician asking, What do you be doing?

605
01:00:45,370 --> 01:00:50,950
And he said, I've been doing this kind of statistical analysis of the way the frequencies occur, and this is what was the formula.

606
01:00:51,400 --> 01:01:00,760
And then physicist. And that's exactly the same formula that we're using to predict energy levels in long atoms like uranium and nuclear.

607
01:01:00,760 --> 01:01:05,860
So these. So it's very difficult to solve the equations for the energy levels, but they have this math.

608
01:01:06,850 --> 01:01:11,650
And the pattern was just such a bizarre one that you just can't believe that to optimise it.

609
01:01:12,040 --> 01:01:17,709
So it's the most exciting progress on the Riemann hypothesis has really been.

610
01:01:17,710 --> 01:01:20,950
That's mainly the notes that Riemann discovered.

611
01:01:20,950 --> 01:01:26,000
Those frequencies share a lot in common that the mathematics that we need to understand them

612
01:01:26,680 --> 01:01:31,900
is actually the same mathematics that we need to understand energy levels in quantum physics,

613
01:01:32,290 --> 01:01:36,460
and those are controlled by all races, eigenvalues and matrices, basically.

614
01:01:36,910 --> 01:01:43,450
So the hunt now really is, is for well, can we find one operator on line one in quantum physics,

615
01:01:43,450 --> 01:01:48,180
which tells you what the energy levels are, which sort of controls the primes energy.

616
01:01:49,120 --> 01:01:55,389
And although we haven't actually come up with what an operator is that's looking at the

617
01:01:55,390 --> 01:02:01,600
primes as almost a physical model has enabled us to discover things about prime citizens,

618
01:02:01,600 --> 01:02:04,210
mathematicians that we've been unable to crack.

619
01:02:04,630 --> 01:02:13,540
So I think it's an exciting time when we're getting this cross-disciplinary interaction, which is helping us to see new ways to look again.

620
01:02:13,690 --> 01:02:17,770
It's coming back to Bulgari. Perhaps we need to ask a different question and you find a new way in.

621
01:02:20,410 --> 01:02:30,150
There's a question of back. Thank you.

622
01:02:30,790 --> 01:02:35,830
So if you think about a sawtooth wave, square wave or something like that,

623
01:02:36,220 --> 01:02:41,770
it's quite clear that you can dramatise it either in terms of an infinite some of its

624
01:02:41,770 --> 01:02:46,810
harmonics like you show or in terms of something like its period in its amplitude.

625
01:02:46,840 --> 01:02:50,380
One of those is actually very simple and the other is very hard.

626
01:02:50,800 --> 01:02:55,780
And I can't help but think that maybe our understanding of the running hypothesis is only

627
01:02:55,780 --> 01:02:59,590
through these harmonics because we haven't seen perhaps the simple way that's missing.

628
01:03:00,100 --> 01:03:05,200
Do you think that's a fair analogy? Oh, I think that is a fair analogy.

629
01:03:05,380 --> 01:03:12,130
And that's what often happens is you can get very boxed in by a particular way of looking at things.

630
01:03:12,130 --> 01:03:18,100
And that's why I think always you need to be open to a new way to look at a subject.

631
01:03:19,120 --> 01:03:27,189
There's a I don't know whether you've seen this psychological experiments where you're showing a video and there are two basketball teams,

632
01:03:27,190 --> 01:03:30,159
one in black, one in white, both possible between each other.

633
01:03:30,160 --> 01:03:37,870
And you're asked to count the number of times the black team passes the ball and you're sold the red herring.

634
01:03:38,170 --> 01:03:42,450
Oh, men and women counted differently and you want to be count in.

635
01:03:42,460 --> 01:03:46,620
You get about 17, you see about 17 and and then experimenting.

636
01:03:46,630 --> 01:03:51,460
Also, did you see the gorilla walk across the street, bang his chest and walk off?

637
01:03:51,670 --> 01:03:59,709
And the first time I saw this video, I did this experiment at the Royal Society know in the Navy at every junction office,

638
01:03:59,710 --> 01:04:04,750
and they rerun the program showing exactly what everybody looked at.

639
01:04:04,870 --> 01:04:11,530
There were a few Tallahassee because they had been concentrating. So I'm going to count how many times the ball guys to the video.

640
01:04:11,890 --> 01:04:22,630
One, two, three, four, five, six, seven. And and I think this is a good reminder that in all of the subjects that we studied, science, mathematics,

641
01:04:22,840 --> 01:04:30,960
that we can get very fixated on a particular way of looking at a subject such that we miss just some obvious gorilla across that.

642
01:04:31,270 --> 01:04:39,220
So I think that's, you know, we have over very senses this we do believe this is the way to unlock the secrets of primes through Raven's work.

643
01:04:40,240 --> 01:04:44,620
But every now and again, somebody comes up with a slightly different perspective on it.

644
01:04:44,940 --> 01:04:53,710
Absolutely. I think that's always the hope that maybe there's a different way to look at where where all of this becomes amazingly simple.

645
01:04:53,800 --> 01:04:57,459
But, you know, we've been looking at this thing for a long time and looking at,

646
01:04:57,460 --> 01:05:03,910
you know, it is the biggest crime and nobody's come up with a different way.

647
01:05:03,910 --> 01:05:08,350
So at the moment, I think we stopped living in Freeman's house, but which I think is a beautiful house.

648
01:05:08,350 --> 01:05:15,370
I think it's a beautiful place to be. But but I think we should always be open to the possibility that somebody might

649
01:05:15,370 --> 01:05:20,559
come from totally that field and show us a new way of looking at things,

650
01:05:20,560 --> 01:05:29,330
which reveals a simplicity. We've just. Let's have a question here and a question here.

651
01:05:31,250 --> 01:05:35,120
You. So there are some other conjecture.

652
01:05:35,220 --> 01:05:41,490
Is that in hindsight going to conjecture is or the dream prior conditions and that if

653
01:05:41,490 --> 01:05:46,950
we assume that the Riemann hypothesis or if we can make more progress about it soon,

654
01:05:47,370 --> 01:05:54,770
do you think proving the Riemann hypothesis, he can be some kind of master key to crack all the other conjectures?

655
01:05:56,700 --> 01:06:02,280
And that's a very good question. And in some ways the answer is probably no,

656
01:06:02,310 --> 01:06:12,240
because very often we think we know how claims are laid out because and and a lot of the mathematics only do I have the beginning of my theorem.

657
01:06:12,240 --> 01:06:15,810
Assume the Riemann hypothesis is true, which is what we all believe.

658
01:06:16,320 --> 01:06:20,940
Then I can make this progress and do things. But similar a lot of problems of our times.

659
01:06:20,940 --> 01:06:25,769
And even if you know the Riemann hypothesis, since they they don't tell you about goldbach conjecture,

660
01:06:25,770 --> 01:06:30,190
for example, some of these conjectures have progress based on the Riemann hypothesis.

661
01:06:30,190 --> 01:06:33,480
So in a way it is the golden key to everything.

662
01:06:35,100 --> 01:06:41,820
In particular, it probably is the golden key to understanding how to crack these codes based on primes.

663
01:06:42,870 --> 01:06:51,660
I mean, I think it's always fun to set up the the the importance of the Riemann hypothesis because it could bring down the Internet.

664
01:06:52,320 --> 01:06:57,180
But actually, you know, if I ask for money and I'll say no, I bet you that when reading a hypothesis,

665
01:06:57,180 --> 01:07:03,630
it proves that it probably won't give rise to a complete collapse of in business.

666
01:07:04,110 --> 01:07:08,459
But again, you know, the Riemann hypothesis we know doesn't tell you anything about well,

667
01:07:08,460 --> 01:07:18,330
it does help some algorithms for cracking numbers, but it won't apply already tell you how to crack numbers into into primes.

668
01:07:18,600 --> 01:07:21,720
But on the other hand, you know, we never know what's the proof will tell us.

669
01:07:21,750 --> 01:07:30,570
And I think that's often why we study these things. This building here in Oxford is named after Andrew Wiles, who proves Fermat's Last Theorem.

670
01:07:30,600 --> 01:07:37,530
Now, Fermat's Last Theorem was a separate equation, actually, and was like the idea was at the end has a solution.

671
01:07:38,370 --> 01:07:41,820
But what we really care about is the mathematics that school generated.

672
01:07:42,240 --> 01:07:47,040
We understood so much deeper about our subject because of the calculus fall back home.

673
01:07:47,340 --> 01:07:51,230
So I don't think anybody is used to suppose that that's not true.

674
01:07:51,240 --> 01:07:58,080
Therefore I'm going to do this. But the mathematics generated was extraordinary, and I suspect that when we eventually prove the Riemann hypothesis,

675
01:07:58,320 --> 01:08:01,990
it won't necessarily be the Riemann hypothesis, which will be the golden key.

676
01:08:02,010 --> 01:08:03,420
But the way we proved it,

677
01:08:03,690 --> 01:08:12,480
which will actually give us far greater insights into things like over 20 primes and maybe even cracking numbers is that prime mover?

678
01:08:15,010 --> 01:08:19,700
You know the question here. Okay.

679
01:08:19,750 --> 01:08:24,400
Of course, that was the same time. Any other questions?

680
01:08:28,600 --> 01:08:33,600
I think that's great. Thank you very much for taking part and everyone is coming later.

681
01:08:33,640 --> 01:08:37,290
You can ask the question in a second to make sure it's not from the vice.

682
01:08:37,290 --> 01:08:39,630
Then thank you very much for being a great audience.