1 00:00:24,710 --> 00:00:29,810 So. Welcome, everyone. I, i work in the mathematics department and occasionally give talks like this. 2 00:00:30,260 --> 00:00:32,600 I'm going to tell you a bit about prime numbers today. 3 00:00:33,500 --> 00:00:43,639 You might well imagine most people know what a prime number is in case you don't, let me just take you through what the definition is. 4 00:00:43,640 --> 00:00:48,860 So prime numbers, something that can't be broken down really into any of the factors. 5 00:00:49,250 --> 00:00:53,719 So that number is like two and three and five and six definitely isn't because 6 00:00:53,720 --> 00:00:58,040 it's three times two and we don't usually consider one to be a prime factor. 7 00:00:58,130 --> 00:01:02,120 A prime number. That's something I'll tell you a little bit about in a moment. 8 00:01:02,510 --> 00:01:07,340 So we start listing these prime numbers. These are what they look like. 9 00:01:07,350 --> 00:01:13,620 They seem to go on a little randomly. They go to three, five, seven, 11, 13, 17, and so on. 10 00:01:14,460 --> 00:01:20,670 And then they go on. They get slightly more rare. But the issue is still the list seems to go on for quite a way. 11 00:01:21,060 --> 00:01:26,850 And in fact, the list goes on forever. And that's something I hope to prove today might be a little bit hard. 12 00:01:26,850 --> 00:01:32,250 So I'll try and take you through through that proof. You might wonder why people are interested in prime numbers. 13 00:01:32,640 --> 00:01:40,040 And that's something I want to give you a sense of today. So if you think about the times, you might have bought something from an Internet company, 14 00:01:40,050 --> 00:01:44,820 perhaps like Amazon or something like that, downloaded something or had your parents do it for you. 15 00:01:45,060 --> 00:01:48,900 It's actually prime numbers that's making all that secure that your parents credit 16 00:01:48,900 --> 00:01:53,190 cards aren't being sort of their numbers aren't being found out by other people. 17 00:01:53,400 --> 00:01:57,150 What makes it all secure is actually prime numbers. I want to say a little bit about that as well. 18 00:01:58,020 --> 00:02:02,460 And people also test out really fast computers trying to find the next big prime number. 19 00:02:02,880 --> 00:02:10,200 This is the largest known prime number at the moment, too, to the 57 million or so it was found earlier this year. 20 00:02:11,070 --> 00:02:15,870 It's got 17 million digits. Okay. So you never run out of prime numbers. 21 00:02:15,870 --> 00:02:19,710 They go on forever. And that's the biggest one we know at the moment. 22 00:02:23,020 --> 00:02:27,280 So let's let's see a little bit about why prime numbers are important. 23 00:02:27,310 --> 00:02:36,459 So if you if I gave you any number like, well, six is the easy one up there you can write six is two times three, oh, three times two. 24 00:02:36,460 --> 00:02:44,350 But that's basically the same thing you can write 24 well, 24 is two times two times two times three. 25 00:02:44,590 --> 00:02:49,840 But every number can be broken down into prime numbers. 26 00:02:49,840 --> 00:02:55,420 Every whole number can be broken down like that. Some of them, like this big number, are actually just prime and that's just it. 27 00:02:55,840 --> 00:02:59,770 And over numbers like six and 24 can be broken down into smaller things. 28 00:03:00,190 --> 00:03:03,390 Do you know how I can get this slide? Is all here still? No. 29 00:03:03,400 --> 00:03:07,330 No one knows. Okay. I was going to use the board as well anyway. 30 00:03:07,330 --> 00:03:16,000 I'll. I'll. You'll be seeing that will find all numbers in fact can be written in this way so everything can be broken down. 31 00:03:16,450 --> 00:03:21,160 And that sort of makes them like the atoms of multiplication, everything. 32 00:03:21,550 --> 00:03:25,430 If you think of this lot like a, a numbers, a molecule really can be broken down. 33 00:03:25,430 --> 00:03:31,209 There's certain way into these atoms and 24 can just be written as two times, two times two times three, 34 00:03:31,210 --> 00:03:36,040 and only in that way, rarely you could rejig the numbers around and put them in a different order. 35 00:03:36,400 --> 00:03:40,060 But really there's just those ways of doing it. 36 00:03:41,860 --> 00:03:46,360 Okay. And this is why we sort of think that, one, it really shouldn't be counted as a prime number. 37 00:03:46,360 --> 00:03:51,820 So if you if you thought one was a prime number, well, then what I just told you would be wrong. 38 00:03:52,480 --> 00:03:55,930 There's you can write six is two times three. 39 00:03:55,930 --> 00:03:59,950 But if one was a prime number you could also Roger is one times two times three. And that would be a different way of doing it. 40 00:04:00,250 --> 00:04:05,170 A One times one, times two times three. So I think I've worked out how to get these balls up. 41 00:04:05,170 --> 00:04:13,690 Let me. That looks promising. Okay. So because of that, you want to say every number can be broken down in a unique way. 42 00:04:13,960 --> 00:04:17,620 And so we don't really want to say one is a prime number. 43 00:04:18,760 --> 00:04:21,969 I've got to stand here doing this. Let's just nothing. 44 00:04:21,970 --> 00:04:26,830 That's far enough for now. So how would you find these out? 45 00:04:26,830 --> 00:04:30,830 Let me give you an example here. You need to find some patterns area. 46 00:04:32,290 --> 00:04:38,710 So if I said how would you break down a number into fact, I'll get this off the board for now. 47 00:04:39,160 --> 00:04:43,600 One seven. 1794. 48 00:04:43,650 --> 00:04:50,760 Okay. How would you break this down into all its numbers that make it up? 49 00:04:51,930 --> 00:04:58,590 And I think you all must know that if that number ends in knots, then it's got a factor of ten in there or a factor of two and a factor of five. 50 00:04:59,130 --> 00:05:05,910 So, you know, if you you can divide this by two and five and you'd get one seven, nine four. 51 00:05:07,440 --> 00:05:15,630 And because this number ends in two, sorry, engine four, you know, it's an even number and it's divisible by two itself. 52 00:05:16,170 --> 00:05:20,430 And if you break that down further, you're going to get eight, nine, seven. 53 00:05:23,920 --> 00:05:27,450 Now, it might not be very apparent to you. So that's divided by two. 54 00:05:30,310 --> 00:05:36,130 That was divided by two and five. What about eight, nine, seven? 55 00:05:36,160 --> 00:05:42,220 Does anyone know what eight, nine, seven might be factored by three goes into it because you can. 56 00:05:42,310 --> 00:05:48,850 Perhaps he's. I'm not sure how he spelled it. Perhaps the easiest thing is to say, well, nine hundreds of multiple of three three goes into the 900. 57 00:05:49,270 --> 00:05:56,320 And this is just 300, 900. So if you divide by three, you get 299. 58 00:05:58,510 --> 00:06:02,700 Now, does anyone know what goes into 299? Perhaps. 59 00:06:02,700 --> 00:06:06,509 Is that a sneak preview for the next ball? They know. But other than that, this is a bit hard. 60 00:06:06,510 --> 00:06:11,180 And you sat there thinking, is that prime or just three go into it? 61 00:06:11,190 --> 00:06:14,580 No, the seven. Well, you'd have to go and check. 62 00:06:14,970 --> 00:06:23,550 And actually it turns out that 299 is 13 times 23, but it take you a while to go and check that. 63 00:06:24,150 --> 00:06:30,540 So in the end, this is 13 times 23 and you've ended up getting all the factors. 64 00:06:30,540 --> 00:06:34,600 We've got a two, a five, a two, a three, a 13 and a 23. 65 00:06:35,370 --> 00:06:41,310 But I hope you can see now that if you're starting to get bigger numbers and obviously doesn't have a two or three that goes in, 66 00:06:41,610 --> 00:06:48,810 it might take you quite a while, even with a calculator to work out just how this is going to affect Horizon there. 67 00:06:48,900 --> 00:06:54,240 Yet that's one of the ways that the actual Internet security is based, 68 00:06:54,240 --> 00:06:58,050 that actually factories in these things can get quite hard if you've got really big numbers. 69 00:06:59,760 --> 00:07:03,600 Anyway, let me let me take you through this proof. I imagine you haven't seen much in the way proof. 70 00:07:03,600 --> 00:07:10,710 So let me go through this slowly and I'll try and give you an example on the next slide to show how this works. 71 00:07:11,760 --> 00:07:18,780 So, so if you don't quite get all this, don't worry. But this is sort of like a proof from several thousand years ago. 72 00:07:18,780 --> 00:07:24,900 You can see a Greek mathematician called Euclid proved in about 300 B.C. He lived in Alexandria, 73 00:07:25,230 --> 00:07:28,590 now in Egypt, and he proved there are actually infinitely many prime numbers. 74 00:07:29,340 --> 00:07:32,850 And he had a very clever way of doing these called proof by contradiction. 75 00:07:33,420 --> 00:07:35,190 So we sort of like say to ourselves, 76 00:07:35,640 --> 00:07:43,650 let's pretend this is false and try and make some conclusions and get to a point where actually we've ended up with some nonsense. 77 00:07:43,650 --> 00:07:47,190 So we know we started with a wrong, wrong starting place. 78 00:07:47,820 --> 00:07:55,080 So what he says is, let's suppose there were just finite many prime numbers that the list stops. 79 00:07:55,380 --> 00:08:01,020 It doesn't go on forever. It actually stops. And if you've only got so many of those prime numbers, you could list them all. 80 00:08:02,370 --> 00:08:06,239 And what is clever idea is if you've just got so many prime numbers like P1, 81 00:08:06,240 --> 00:08:10,320 P2 up to P.K., these are all the prime numbers, Euclid fault there might be. 82 00:08:10,860 --> 00:08:15,630 What he did is he multiplied them all together and added one OC. 83 00:08:15,750 --> 00:08:22,350 And why he did that is because if you multiply all those numbers together and add one, you've definitely got a number. 84 00:08:22,350 --> 00:08:26,550 Now that none of the other numbers that you had divided it. 85 00:08:26,790 --> 00:08:31,350 Okay, if you divide them main, they all leave remainder one, none of them go in. 86 00:08:32,220 --> 00:08:39,780 But that new number, that really big number he's probably created and doesn't have any of your known prime numbers as factors. 87 00:08:40,470 --> 00:08:44,130 So the only possibility is now over. It's prime. 88 00:08:44,790 --> 00:08:48,930 Or if you went and looked at its prime factors, they wouldn't be on your list. 89 00:08:49,380 --> 00:08:54,840 You've deliberately found something that's new, gives you a new prime number one way or another. 90 00:08:55,560 --> 00:09:02,250 So just in case that might seem a little bit confusing, let's try and do it with an example and see what Euclid's method would have done. 91 00:09:03,810 --> 00:09:08,580 So I'm like, Let's say we were really quite lazy and we only fault there were two prime numbers, 92 00:09:08,580 --> 00:09:11,700 two and five, which is a very good effort, at least in them all. 93 00:09:11,700 --> 00:09:21,180 But let's say we started with that. Then what Euclid would say is take you two and you five, multiply them together, add one, and that gives us 11. 94 00:09:21,900 --> 00:09:29,310 Now, either this new number, 11 is prime, or if we factories 11, we get some numbers that aren't two or five. 95 00:09:30,120 --> 00:09:34,440 So in this case, here we get 11 and that's a new prime. So we've got one more on our list. 96 00:09:35,460 --> 00:09:41,310 So we might say, okay, maybe we've got them all. Now let's take two and five and 11. 97 00:09:41,340 --> 00:09:50,850 I think I've got them all. Well, Euclid would say multiply them all together, add one and you got 111 and it isn't prime actually 111, 98 00:09:51,480 --> 00:09:56,730 but it's prime factors which are three and 37 and new ones not on our list. 99 00:09:57,270 --> 00:10:01,739 So if, however, we felt fault, we'd got all the prime numbers, he had a way of saying, 100 00:10:01,740 --> 00:10:05,850 multiply them all together, add one, and I'm guaranteed to get you a new prime number. 101 00:10:06,330 --> 00:10:10,020 So that means you can never list them all. The list must go on forever. 102 00:10:11,810 --> 00:10:15,530 But anyway, the reassuring thing is we're never going to run out of prime numbers. 103 00:10:15,860 --> 00:10:19,880 You can always find bigger and bigger and larger prime numbers. 104 00:10:20,900 --> 00:10:23,990 And there's nothing images actually from in this museum. 105 00:10:25,250 --> 00:10:29,060 This is you played our statue of Euclid. I guess no one really quite knows what he looks like. 106 00:10:29,870 --> 00:10:35,960 And this is a statue of him in the museum. He wrote very famous books called The Elements. 107 00:10:36,380 --> 00:10:40,940 And there's been more editions of The Element than probably any book except the Bible. 108 00:10:40,970 --> 00:10:47,450 I mean, this was really the mass book of the maps books until really the start of the 20th century. 109 00:10:48,260 --> 00:10:49,910 So a very famous Greek mathematician. 110 00:10:53,650 --> 00:11:02,530 So if I if I gave you some examples a bit like this again and said, which of these a prime how easy would some of them be to find? 111 00:11:02,540 --> 00:11:07,450 Does anyone want to make any cross a few off my list and say such and such isn't prime going. 112 00:11:10,180 --> 00:11:14,220 Okay. And we know that because what what factor of using. Okay. 113 00:11:14,260 --> 00:11:20,380 So even if there's any even ones that we know they're not going to be prime anyone else cross any of the areas of. 114 00:11:20,420 --> 00:11:25,149 Yeah. The second one because. Okay. 115 00:11:25,150 --> 00:11:30,700 So you know if something ends in five or not, it's divisible by five in any one spot. 116 00:11:30,700 --> 00:11:34,410 Any of the others. We are going. 117 00:11:38,200 --> 00:11:41,440 Oh, I'd love to have a think about that. 118 00:11:41,760 --> 00:11:45,880 It depends. What? Whether 273 is divisible by seven. Yes, it is. It is divisible by seven. 119 00:11:46,120 --> 00:11:50,290 Very small. And it's also divisible by three. But good, good sport. 120 00:11:51,040 --> 00:11:55,540 What about any of the others? Yep. 121 00:11:58,510 --> 00:12:03,060 Uh, I don't think so. Why do you think it is? 122 00:12:05,220 --> 00:12:08,310 Uh, no, definitely isn't, I'm afraid. 123 00:12:08,430 --> 00:12:13,260 Isn't divisible by three. Can anyone spot the fall four and what that might be divisible by? 124 00:12:15,940 --> 00:12:19,510 Anyone. Because he's got a certain pattern to it doesn't it. 125 00:12:19,810 --> 00:12:24,880 337337. And they won't think of you doing very well. 126 00:12:30,900 --> 00:12:35,680 Sorry. 14 no football. 127 00:12:35,850 --> 00:12:44,250 All the lost ones are old so 14 could divided also be an even number now 337 you see divides the four four. 128 00:12:44,850 --> 00:12:47,910 Can anyone see how many times 337 goes into the fourth one. 129 00:12:51,670 --> 00:12:54,979 No. So 1001 times. 130 00:12:54,980 --> 00:12:58,969 It's just that's why it sort of repeated like that. It's 337 times. 131 00:12:58,970 --> 00:13:04,790 2001 is the fourth one. But you can see they're getting a bit harder and. 132 00:13:06,590 --> 00:13:09,700 Here are some tricks for simple things. 133 00:13:09,710 --> 00:13:15,170 Obviously, it's fairly easy to work out whether something is divisible by two and it's not too hard for four, 134 00:13:15,170 --> 00:13:22,370 and eight over five is also fairly easy, mainly because we just use decimal and that's base ten. 135 00:13:22,370 --> 00:13:34,790 So two and five factors of tightness that makes a bit easier if you want to pull the numbers in all the digits in a number like for example, 136 00:13:35,150 --> 00:13:40,040 2732, one few other two and seven and three and two and one. 137 00:13:40,040 --> 00:13:45,949 You get what, nine, 12, 14, 15 and because 15 is divisible by three, so is that number. 138 00:13:45,950 --> 00:13:50,870 That's a trick you might not have seen before if you had to pull the digits and it's 139 00:13:50,870 --> 00:13:55,189 divisible by three that so the number would have been and the same worked for nine as well. 140 00:13:55,190 --> 00:14:00,440 If you add up all the numbers and it's divisible by nine the sum, so would the original number. 141 00:14:00,440 --> 00:14:04,400 So that number isn't divisible by nine because 15 is not divisible by nine. 142 00:14:05,090 --> 00:14:15,260 But these are just tricks for small numbers. If you wanted to work out some of the others, they're getting a bit hard as it happens. 143 00:14:17,210 --> 00:14:22,250 100 and fall 7 to 9 is prime. It's quite a large prime. 144 00:14:22,280 --> 00:14:25,340 It would take you quite a while to work this out. 145 00:14:25,670 --> 00:14:30,829 This other big number, though, is actually not prime, 146 00:14:30,830 --> 00:14:39,410 but you'd have to try all the prime numbers until eventually you got to 331 to find a factor that's going to take you quite a while. 147 00:14:39,800 --> 00:14:47,540 You'd have to try 66 prime numbers before you actually got to the 67th prime number, and that's going to take you a long time. 148 00:14:47,900 --> 00:14:53,780 So find verifying that's prime and showing that isn't would take quite a while and there were only six digits long those numbers. 149 00:14:55,700 --> 00:15:03,160 Okay. So this is what I'm sort of getting out wave with some of these aspects. 150 00:15:04,540 --> 00:15:08,810 If you're going to guarantee some of these pricing, you basically have to go to the square root. 151 00:15:08,830 --> 00:15:15,100 I imagine you've come across ideas of square roots. So the square root of four is two, square root of nine is three. 152 00:15:15,370 --> 00:15:22,810 You'd have to go up to the square root checking all the prime numbers, and if none of them divided it, then it was prime. 153 00:15:23,410 --> 00:15:35,920 So if you wanted to check the 104,729 was prime, you got to try every prime number, really quite a few of them, 66 until you get up to 323. 154 00:15:36,310 --> 00:15:41,680 And then by that point, you've actually checked his prime. But that's really quite a hard thing to do. 155 00:15:42,430 --> 00:15:53,530 So actually working out in general, this sort of factorisation is quite hard even for computers once the numbers start getting quite big. 156 00:15:54,310 --> 00:15:58,390 So that's really why we use it in security. 157 00:15:59,840 --> 00:16:06,140 Here's a novel way, a clever way. Do it due to a Frenchman man from the 17th century fermor. 158 00:16:07,670 --> 00:16:16,070 So if a P is a prime, an end is a whole number, then the prime number will divide. 159 00:16:16,370 --> 00:16:20,989 And the P minus sign. That's a theorem that you might meet at university if you studied maths there, 160 00:16:20,990 --> 00:16:26,780 but we're only really interested in it from a point of view of helping test prime numbers. 161 00:16:27,350 --> 00:16:32,629 So if we just check it here, what his theorem says is that for something like five, 162 00:16:32,630 --> 00:16:38,360 which is definitely five, definitely prime five divides two to the power of five minus two. 163 00:16:39,080 --> 00:16:47,600 So two to the power, five minus two is 30 to -2 and we get 30 and five also divides three to the five minus three 164 00:16:47,600 --> 00:16:54,470 because that's 240 and five will also divide fall to the five minus four because that's 1020. 165 00:16:55,490 --> 00:16:58,220 And so if you turn it on its head and say, well, 166 00:16:58,580 --> 00:17:07,520 suppose I had a number that didn't this theorem didn't hold for then I knew it wasn't prime could be useful the oval way round. 167 00:17:07,520 --> 00:17:15,020 Well having to check all these factors you could do one possibly big calculation, but one calculation and get it right. 168 00:17:15,770 --> 00:17:22,070 And so you could say to yourself, well, if P is prime, it's going to divide two to the P minus two. 169 00:17:22,940 --> 00:17:33,110 So if I have a number that isn't prime or if I have a number N and it doesn't divide two to the N minus two, then I won't be prime. 170 00:17:33,110 --> 00:17:35,690 And sometimes that's an easy thing to work through. 171 00:17:36,020 --> 00:17:42,260 Then when you've got a large number and you're trying to work out all its possible factors, so that could be a way of crossing it off. 172 00:17:43,310 --> 00:17:48,020 The trouble is there are some nasty numbers out there that still satisfy all these tests. 173 00:17:48,020 --> 00:17:53,510 So we thought we'd call these things pseudo primes, and they do this. 174 00:17:53,630 --> 00:18:05,930 They satisfy all this without being prime themselves. It won't be very obvious, but 561 is one of these numbers, so 561 satisfies all. 175 00:18:07,070 --> 00:18:16,370 This Theorem 561 would always divide added to the 561 minus sign, but it isn't itself prime. 176 00:18:16,370 --> 00:18:23,509 So there are some annoying numbers out there that sort of like even if you checked with this, you wouldn't be 100% sure they were prime. 177 00:18:23,510 --> 00:18:31,339 You just you'd think they were probably prime. But there are annoyingly 100 there are infinitely many of these false primes out there. 178 00:18:31,340 --> 00:18:35,960 So actually working out whether some of these prime are not can be quite, quite hard. 179 00:18:36,320 --> 00:18:42,770 There are ways of quickly showing something isn't, but there are these false primes out there. 180 00:18:44,060 --> 00:18:50,240 Let me change topic a little bit here, first of all. So that's so picture of wiles. 181 00:18:51,320 --> 00:18:53,090 This is on true now? I'm afraid so. 182 00:18:53,240 --> 00:18:58,520 So the new maps built in which you're actually going to I think after this or some of you it's called the Andrew Wiles building. 183 00:18:59,240 --> 00:19:04,550 And the Andrew Wiles building was named unsurprisingly after Andrew Wiles, who is here doing Master Oxford. 184 00:19:04,830 --> 00:19:13,910 He's the man who proved Fermat's Last Theorem, which was one of the really famous theorems of mathematics that been unproved for 350 years. 185 00:19:14,300 --> 00:19:22,910 And he came along and proved it in the nineties. But so you may well be seeing the new Andrew Wiles building after this. 186 00:19:23,110 --> 00:19:26,450 Fermat was an important mathematician in the 17th century. 187 00:19:27,970 --> 00:19:34,520 Let's change topic a little bit now and think to ourselves, how is this going to be useful? 188 00:19:34,540 --> 00:19:41,800 Well, I'm talking about in the real world because you might be thinking prime numbers are nice enough, but what point are they? 189 00:19:43,180 --> 00:19:45,420 Well, actually, you use them a lot without really knowing. 190 00:19:45,430 --> 00:19:52,060 It's like I say, if ever you buy anything off the Internet, then you basically are using prime numbers. 191 00:19:53,650 --> 00:20:02,050 So what's the setup in the Internet? You've got yourself your computer and you sort of like on the Internet with a Internet company. 192 00:20:02,200 --> 00:20:11,260 How are they going to be able to get from you, your or your parents credit card number to them without anyone that saying, 193 00:20:11,860 --> 00:20:18,040 you know, intercepting the communication or anyone who would like to steal your computer or something like that. 194 00:20:18,250 --> 00:20:21,790 How can you be sure that your credit card number will be safe? 195 00:20:24,210 --> 00:20:29,850 And you got to remember, there are millions and millions of people out there using the Internet all the time to buy things. 196 00:20:31,130 --> 00:20:37,810 Well, there are always all sorts of problems with communications in front of you if you want to and code sorry. 197 00:20:37,820 --> 00:20:42,740 If you want to read a good book on this, there's a book by Simon Singh. 198 00:20:45,680 --> 00:20:57,510 Just call the code book. It's a very good read for finding out about how people used to communicate in secret over the years. 199 00:20:57,930 --> 00:21:01,739 It's not it's not hugely mathematical, but there's. So it's a very good read. 200 00:21:01,740 --> 00:21:04,920 And so some of this detail will be in that book as well. 201 00:21:05,850 --> 00:21:13,319 So how might you do this? If you think to yourself, I'd like to keep a secret, but I want to, you know, share it with a few friends, 202 00:21:13,320 --> 00:21:18,720 but only those friends so that it really is secret amongst us, but no one else. 203 00:21:19,740 --> 00:21:25,830 Well, they used to do this many, many years ago by just using a very simple cipher. 204 00:21:26,490 --> 00:21:29,880 What you might do is you've got this message you want to send. 205 00:21:30,630 --> 00:21:35,250 What you could do is just change every letter in some way that you was agreed. 206 00:21:35,460 --> 00:21:39,990 Okay. You and all your friends would know that this was how you were going to communicate. 207 00:21:39,990 --> 00:21:46,049 You were going to scramble your message by taking a word and swapping letters, for example. 208 00:21:46,050 --> 00:21:55,550 Let me give you an example here. So what you do is at the top here, you've got yourself what's called the key. 209 00:21:56,120 --> 00:21:59,180 Whenever you see a nay, you're going to replace it by a G. 210 00:21:59,540 --> 00:22:04,610 Whenever you see a B, you're going to replace it by a P. Whenever you see your name, you're going to replace it by a Q. 211 00:22:05,270 --> 00:22:09,290 So if I get a phrase like Oxford Mathematics, if I've done this right. 212 00:22:09,650 --> 00:22:14,960 O became L and X became W and F became N and so on. 213 00:22:15,560 --> 00:22:22,880 But if you sort of like manage to, oh, if you lost this message or your friends lost it. 214 00:22:23,240 --> 00:22:25,610 Anyone just picking this up, would it know what they said? 215 00:22:26,360 --> 00:22:32,180 Okay, this would be a scrambled message and they wouldn't know anything about what you've been trying to say to one another. 216 00:22:33,290 --> 00:22:40,850 The trouble is, once you start saying a lot to one another over, people could stop making guesses as to what the how it got scrambled. 217 00:22:41,630 --> 00:22:48,830 Because the I mean, there's all sorts of problems here. One is, how do you carefully get this private key to all your friends? 218 00:22:49,550 --> 00:22:56,930 And then if you keep communicating enough, people can start making guesses as to what you might have been writing. 219 00:22:57,800 --> 00:23:10,340 So if I had a page, you know, quite a long bit of text and I used my key to scramble it all, sent it to you, but you accidentally got intercepted. 220 00:23:11,460 --> 00:23:18,020 What what letter do you think might appear most in the scrambled, uh, scrambled work? 221 00:23:18,020 --> 00:23:23,700 Can anyone make a guess? If I was using this copier, what do you think would appear most? 222 00:23:24,510 --> 00:23:29,920 G. G. You're going for G because. H on the ray. 223 00:23:30,290 --> 00:23:36,309 He's a very common light. That's a good suggestion. There would be a lot of GS, but it wouldn't be the most common one. 224 00:23:36,310 --> 00:23:38,799 Does anyone know? It would be J? 225 00:23:38,800 --> 00:23:45,700 But there would be a lot of GS, but the probably be most j's because E's the most common letter in the English alphabet. 226 00:23:45,730 --> 00:23:50,860 Does anyone know what the second most common is? No. 227 00:23:51,010 --> 00:23:54,500 Afraid not. If we could be a while. I suppose we have to have this. It's gone. 228 00:23:55,600 --> 00:23:59,180 No, no, it's actually t. T is the next most common one. 229 00:23:59,720 --> 00:24:03,290 But you can see you're already guessing the way people are. 230 00:24:03,290 --> 00:24:07,130 If you saw a lot of JS, you'd be thinking, maybe that came from me. 231 00:24:07,580 --> 00:24:14,719 Maybe that came from T. And then if you saw, like saw a lot of things that were maybe T something e you might be thinking to yourself, 232 00:24:14,720 --> 00:24:17,870 that's perhaps an H that was the something between them. 233 00:24:18,500 --> 00:24:28,190 This is what Phumzile call frequency analysis. So here are the how common the English letters are. 234 00:24:28,760 --> 00:24:37,459 So you can see E appears most commonly at 12.7% of the time on t is the next most common one at 9.1. 235 00:24:37,460 --> 00:24:41,750 But as common as someone said, and this is also pretty common as well. 236 00:24:42,470 --> 00:24:48,049 So there's you could sort of once you add a lot of text, you could start making educated guesses. 237 00:24:48,050 --> 00:24:55,070 You wouldn't want to just go in randomly and start thinking this might be that and this might be that, but you could do some frequency analysis. 238 00:24:55,760 --> 00:25:02,600 So it was actually Islamic mathematicians around 800 A.D. that actually started being able to break down this code. 239 00:25:02,600 --> 00:25:07,280 So you can't just use a set, especially in this day of modern computers. 240 00:25:07,670 --> 00:25:12,860 You can't just sort like use a SCI for the way they used to do back in Roman times or something like that. 241 00:25:12,860 --> 00:25:17,840 It would just be much too easy to crack. So you don't have to be a bit smarter. 242 00:25:18,860 --> 00:25:22,309 Also, if you were sort of like an Internet company, perhaps someone like Amazon or something like that, 243 00:25:22,310 --> 00:25:28,100 they've got millions and millions of customers, even if they came up with something a bit better than this. 244 00:25:28,640 --> 00:25:34,850 Are they really going to have to send all this to all the customers around the world and be sure that 245 00:25:34,850 --> 00:25:39,380 everyone sort of like taking good care of it and not leaving it on the bus and various things like that. 246 00:25:39,740 --> 00:25:47,560 They're really going to have to come up with something better than that. So this is this has always been an issue of what I call private keys. 247 00:25:47,680 --> 00:25:58,420 How do you get the key to your friends? Because if if maybe one of your friends isn't not trustworthy or they lose it all, it just gets intercepted. 248 00:25:58,840 --> 00:26:04,149 All your enemies. Because we might be talking about this at wartime. All your enemies can now have it. 249 00:26:04,150 --> 00:26:08,410 And these are either competitor companies or their enemies at war time. 250 00:26:08,410 --> 00:26:17,860 Security is a major issue. You know, we've we've, you know, personal details with secrets, with things like that could be majorly important. 251 00:26:18,520 --> 00:26:24,380 This information gets kept secret. So these are guys. 252 00:26:25,640 --> 00:26:33,170 So we've asked Jemmy and Adam, and they gave their initials to what is basically the most used bit of mass really, I guess in the world today. 253 00:26:33,920 --> 00:26:38,660 It's a mass or it's a large part of the mathematics behind Internet security. 254 00:26:39,290 --> 00:26:47,000 If you give over your credit card details, send it on to some Internet company, it would be they sell some version of this. 255 00:26:47,210 --> 00:26:49,010 That's basically keeping it a secret. 256 00:26:49,880 --> 00:26:56,630 It's actually it's reasonably new mass because this came in 1977, which I guess might seem a long time ago to you people. 257 00:26:56,630 --> 00:27:01,360 But yeah, I remember 1977, and that's quite new for mathematics. 258 00:27:01,380 --> 00:27:08,630 1977, they came up with this idea. It uses what I told you about earlier Fermat's Little Theorem. 259 00:27:09,170 --> 00:27:13,580 So it's 17th century maths and the Chinese remainder theorem, which is even older maths. 260 00:27:14,240 --> 00:27:18,139 So the actual maths behind this, you could explain to Fermat. I need to understand it. 261 00:27:18,140 --> 00:27:26,450 Fine. What he'd be astonished about is how you've got to know about prime numbers that are like hundreds and millions of digits long. 262 00:27:26,850 --> 00:27:29,190 He'd have no idea about the modern computer age. 263 00:27:29,190 --> 00:27:35,000 He wouldn't know how you'd be able to come up with a prime number of 17 million digits, like we mentioned earlier. 264 00:27:36,800 --> 00:27:40,940 So I'm going to tell you a little bit about how prime numbers are used in this. 265 00:27:42,030 --> 00:27:45,870 So this is sort of the we don't really want to use a private key system. 266 00:27:46,380 --> 00:27:57,300 We want to use a public key system. It's a bit better. So what the Internet company does is tell all its customers or really the customers computers, 267 00:27:57,660 --> 00:28:07,170 how they're going to be able to scramble their message so that you can send on your credit card details in a careful way. 268 00:28:07,890 --> 00:28:11,549 You won't be able to unscramble them, but you will be able to scramble them. 269 00:28:11,550 --> 00:28:19,860 And that's quite, quite so. So what they do, the Internet company sends to your machine something called the public key. 270 00:28:21,060 --> 00:28:26,550 They don't mind if it gets lost the Internet company, because all it is is a means of scrambling. 271 00:28:26,640 --> 00:28:30,420 Okay. Then your computer gets it. 272 00:28:30,750 --> 00:28:34,950 You go and you buy what you want on the Internet and you send your details on. 273 00:28:35,190 --> 00:28:41,310 And only the Internet company knows how to scramble it, unscramble it, decrypt it is a fancy word. 274 00:28:42,700 --> 00:28:52,390 Okay. And what makes this secure and what makes this basically mean that no one's going to be able to break it is if you multiply numbers together, 275 00:28:52,870 --> 00:28:57,790 that's easy. Any computer can do it quite quickly, even if the numbers are huge. 276 00:28:58,360 --> 00:29:02,800 But if you're going to have to raise it, then it takes a long, long time. 277 00:29:02,830 --> 00:29:09,430 Okay. So if I gave a computer a very large number and said, please go and work out its factors, 278 00:29:09,730 --> 00:29:11,980 especially if there were many of them and the factors were big, 279 00:29:12,280 --> 00:29:19,360 it might sit there for ages and ages and years and years and in fact, millennia and millennia before we actually came up with the answer. 280 00:29:19,840 --> 00:29:25,840 So that's why the system secure. So if you multiply two large numbers together, it's easy. 281 00:29:26,320 --> 00:29:32,830 If you then gave the computer that product and said find them, then it might sit there working for a long, long time. 282 00:29:33,500 --> 00:29:40,780 And this is what's called a one way function where it's very easy to go one way, an incredibly hard, at least computation wise to come backwards. 283 00:29:42,970 --> 00:29:44,920 So let me give you a few more details here. 284 00:29:46,690 --> 00:29:58,060 So what what goes on in your computer is the Internet company gives you all computer, just two numbers and an E. 285 00:29:58,960 --> 00:30:03,700 And actually what goes on in the computer is very simple, is called the encryption power. 286 00:30:04,330 --> 00:30:08,170 And the number is huge. It might be like a thousand digits long. 287 00:30:09,920 --> 00:30:14,700 And it's only got two factors. He's got two prime numbers as its factors. 288 00:30:16,850 --> 00:30:25,400 All this system is still a cipher. It's like taking those 26 letters we had before and moving them around and rearranging them as 26. 289 00:30:25,610 --> 00:30:28,760 But we're not using 26 anymore. We're now using N. 290 00:30:29,660 --> 00:30:33,830 And so N is about ten to the power of a thousand, a huge, huge number. 291 00:30:34,490 --> 00:30:36,920 And so if you thought to yourself, you know, where am I? 292 00:30:36,920 --> 00:30:42,649 E's on my TS amongst these ten to the thousand numbers, you just can't do frequency analysis anymore. 293 00:30:42,650 --> 00:30:46,520 It's just too, too big. You're going to have to find other ways of doing it. 294 00:30:46,940 --> 00:30:51,680 So it's still a cipher, but we're using a number much, much bigger than 26 now. 295 00:30:55,600 --> 00:31:00,940 And all your computer does. I mean, it might be explained a bit more. 296 00:31:00,940 --> 00:31:05,560 Is something called clockwork arithmetic or modular arithmetic that I've just explained in different ways here. 297 00:31:07,580 --> 00:31:16,070 You want to send them your credit card details? Let's say this is some number and and I'm probably going to be less than nine. 298 00:31:16,190 --> 00:31:20,179 But if not, you could send it by beat your message. So let's say M is bigger. 299 00:31:20,180 --> 00:31:30,020 That is less than. All your computer does is take that number E that the Internet company gave it. 300 00:31:31,070 --> 00:31:38,180 Raise it to the power e divide by the number n and you get some remainder. 301 00:31:38,480 --> 00:31:41,870 It goes so many times does and and that leaves some remainder. 302 00:31:42,200 --> 00:31:49,909 And the remainder is just what gets sent off. So it's very big numbers are involved, but the actual maths is really quite simple. 303 00:31:49,910 --> 00:31:54,620 You raise it to the power e you divide by N and whatever the remainder is you 304 00:31:54,620 --> 00:31:59,449 send on this is now your scrambled message and if someone intercepted it, 305 00:31:59,450 --> 00:32:02,510 it would just be gobbledegook to them. They wouldn't know anything about it. 306 00:32:04,610 --> 00:32:08,570 Maybe there were people there thinking I could break that. That would be easy enough to break. 307 00:32:10,040 --> 00:32:13,340 So let's talk about cracking it. How would you crack this? 308 00:32:13,970 --> 00:32:24,530 Let me actually let me go back. One slide here. If you've got any public key system, what you have told people how to do is encrypt. 309 00:32:25,100 --> 00:32:28,340 That's a bit like giving someone half of an English French dictionary. 310 00:32:28,610 --> 00:32:32,540 Okay, let's say I gave you the English French half of the dictionary. 311 00:32:33,230 --> 00:32:37,010 And I said to you, What does this mean? What does your de mean? 312 00:32:37,340 --> 00:32:43,760 And Thursday. Okay. How long do you think it would take you not knowing that and we've only the English French. 313 00:32:43,790 --> 00:32:47,930 But what would you have to do to work out? Jeopardy meant Thursday. 314 00:32:49,220 --> 00:32:55,730 Why would it take you a while? Yeah. 315 00:32:55,750 --> 00:32:58,670 You'd have to just keep looking. You wouldn't know where to start. 316 00:32:58,670 --> 00:33:02,840 But in principle, if you sat there and sat there and sat there and eventually got to TS, 317 00:33:03,320 --> 00:33:06,440 then you'd work out that Jody met Thursday but would tell you ages. 318 00:33:07,280 --> 00:33:16,640 So if I gave you the English French half of the dictionary and said, make the other half, you could tell you a long, long time. 319 00:33:17,030 --> 00:33:24,710 But if someone tells you how to encrypt or change something into a different language, you always have in principle a way of getting it back. 320 00:33:25,070 --> 00:33:28,340 But that's going to have to take you a long, long time for this to be secure. 321 00:33:28,340 --> 00:33:32,510 And it would take you a long, long time because you'd have to sort of like take in words. 322 00:33:32,960 --> 00:33:36,530 All I know is I've been turned to add numbers and then you'd have to work out 323 00:33:36,530 --> 00:33:40,850 all these calculations and then sort of alphabetise the other half to get back. 324 00:33:41,270 --> 00:33:49,280 That's going to take for ages. Another thing you might try and do is just actually work out what the factors of that end is. 325 00:33:49,730 --> 00:33:53,090 The end is this thousand digit number somewhere in your computer. 326 00:33:53,610 --> 00:33:57,410 If you can go and work how those factors are. You can break the code. 327 00:33:57,920 --> 00:34:00,950 But that's going to take you pretty much forever. Okay. 328 00:34:00,980 --> 00:34:06,350 And the Internet company tomorrow or probably even later this afternoon, are going to be using different prime numbers. 329 00:34:06,800 --> 00:34:10,940 So you don't have the rest of the universe to sort of like work this through. 330 00:34:11,180 --> 00:34:16,280 You've only got sort of like perhaps the next ten, 15 minutes before another set of numbers are used. 331 00:34:17,450 --> 00:34:26,029 So it's pretty secure. At the offer end, the the Internet company has got what matters. 332 00:34:26,030 --> 00:34:31,880 They've got the factors. Then they know those. And because of that, they can work out the decryption power. 333 00:34:32,390 --> 00:34:36,680 So when you've sent on your scrambled message, see? Okay, here. 334 00:34:37,310 --> 00:34:47,719 Then at the other hand, they just raise it to a decryption power d known only to then divide again by N and as if by magic. 335 00:34:47,720 --> 00:34:52,160 But it's just Fermat's Last Theorem and the Chinese remainder theorem. 336 00:34:52,340 --> 00:34:59,600 When you divide by an impulse out, they now know your credit card details what you want to buy, where they need to send it. 337 00:34:59,630 --> 00:35:05,140 All the information comes out, so it's really just quite simple maths that is going on. 338 00:35:06,850 --> 00:35:17,850 Okay. So I can say here, if you wanted to go and try and crack this, then you could do this by trying to do all the scrambling, 339 00:35:18,060 --> 00:35:20,880 rearrange the scrambling, and make the other half of the dictionary. 340 00:35:21,150 --> 00:35:27,230 That would take forever, pretty much all you could try to rise in this number, and that's pretty much going to take forever as well. 341 00:35:27,240 --> 00:35:32,799 So it's a very, very secure system. Okay. 342 00:35:32,800 --> 00:35:40,120 Let me sort of like draw this to a close a bit by talking a little bit about how the primes are distributed. 343 00:35:41,260 --> 00:35:45,610 So I hope you now at least see that primes can be really quite useful. 344 00:35:46,890 --> 00:35:51,660 They're funnier primes and it's very easy to ask very hard questions about them. 345 00:35:52,380 --> 00:35:57,240 You can do all sorts of things. Like if you look at if you look at this list, 346 00:35:59,100 --> 00:36:07,490 you can see you've got three and five next to an end of a five and seven, 11 and 13, 17, 17 and 1929 and 31. 347 00:36:07,500 --> 00:36:12,840 These are what are called twin primes, apart from two no number, that's even as prime. 348 00:36:13,860 --> 00:36:20,490 So all primes are old apart from two. And this means that the nearest two of them could be together is just a difference of two. 349 00:36:21,090 --> 00:36:26,280 And you can see quite a few examples here, 29 and 31, 71 and 73. 350 00:36:26,640 --> 00:36:33,150 And these are called twin primes. And people want to know whether or not they're infinitely many twin primes. 351 00:36:33,150 --> 00:36:38,790 And no one knows. No one's close to knowing, really at the moment there's been progress with that problem. 352 00:36:38,790 --> 00:36:43,799 Just this year, a lot of mathematicians got excited about various results related to that, 353 00:36:43,800 --> 00:36:49,200 but no one actually knows whether there are infinite many, many primes, just two apart. 354 00:36:49,500 --> 00:36:53,430 So it's very easy to ask very hard questions about these sorts of things. 355 00:36:54,240 --> 00:36:59,970 One thing we do sort of understand, though, is just how frequent they are because they might look a bit random, 356 00:37:00,450 --> 00:37:04,830 but long term we can actually get a fairly good sense of how how often they pop up. 357 00:37:05,310 --> 00:37:11,690 And that's something called the Prime Number Theorem. Okay. 358 00:37:12,480 --> 00:37:18,320 And my guess is you might not have met this idea before, so let me just tell you what a factorial is. 359 00:37:19,190 --> 00:37:24,650 You've probably seen it on your calculator. Uh, is this an and then an exclamation mark? 360 00:37:25,110 --> 00:37:31,420 And if you've never been sure what it was, it just multiplies all the numbers up to it together. 361 00:37:31,430 --> 00:37:35,180 So three factorial is one times. Two times three, which is six. 362 00:37:35,840 --> 00:37:42,829 Four factorial is one times two. Three times four, which is 24 and five factorial c is 126. 363 00:37:42,830 --> 00:37:51,680 Factorial 720. Why am I mentioned in this is because you can use this idea to show there are quite large gaps amongst the primes. 364 00:37:52,700 --> 00:37:59,630 If I said to you, find me 100 numbers, one after another that none of them are prime. 365 00:38:00,190 --> 00:38:04,570 I'm not. You might be thinking to yourself, that's impossible. Around 100 numbers, one after another. 366 00:38:04,850 --> 00:38:08,150 They aren't prime, but there are gaps like that in amongst the primes. 367 00:38:08,570 --> 00:38:12,440 And you can actually see that using this idea. Okay. 368 00:38:12,440 --> 00:38:16,400 So there are large gaps in the prime, in the primes, as large as you want them to be. 369 00:38:18,070 --> 00:38:28,660 If I if I take an factorial plus two and you're some number two was in that product ten factorial. 370 00:38:29,860 --> 00:38:34,720 So two divides N factorial and factorial is one times two all the way up to end. 371 00:38:35,410 --> 00:38:38,800 And if I add to this number, still going to be divisible by two. 372 00:38:39,610 --> 00:38:46,810 If you look at the next number, if any is bigger than three, then three will divide and factorial. 373 00:38:47,710 --> 00:38:51,910 Okay. It'll also divide three. So the next number is divisible by three. 374 00:38:52,870 --> 00:38:57,910 And if it's bigger than four or five or more than four will be in four factorial. 375 00:38:57,940 --> 00:39:02,800 Go into four factorial and also be divisible by four. So the first number's divisible by two. 376 00:39:02,830 --> 00:39:08,200 The next one by three. The next one by four. If I choose n as large as I want it to be. 377 00:39:08,470 --> 00:39:12,100 Here is a list of numbers. One after the over. I have no primes. 378 00:39:12,760 --> 00:39:27,130 So, for example, here, five factorial plus 222 is divisible by 223 is divisible by 320 fours, divisible by four, 125 divisible by five. 379 00:39:27,880 --> 00:39:33,550 So if I used N is 101, I'd actually have 100 numbers one after ANOVA, none of which are prime. 380 00:39:33,550 --> 00:39:37,600 So there's gaps as large as you want in amongst the prime numbers. 381 00:39:38,170 --> 00:39:42,820 But at the same time, we sometimes think that infinitely often there's two just next door to one another. 382 00:39:43,420 --> 00:39:49,090 So you get you get odd sort of like spurts of prime numbers and then huge gaps. 383 00:39:52,200 --> 00:39:57,770 Here's another thing you might have seen on your calculator, but perhaps never quite know what it was. 384 00:39:57,790 --> 00:40:06,600 So this is called the Aladdin function. Okay. Which is a I'm not going to use that much here, but Aladdin is basically stands for natural logarithm. 385 00:40:07,920 --> 00:40:12,420 You can work it out on your calculator. That's a graph of what it looks like. 386 00:40:12,420 --> 00:40:16,360 Aladdin of One is not a line of larger numbers. 387 00:40:16,380 --> 00:40:19,830 Aladdin of a number between two and three is one. 388 00:40:21,480 --> 00:40:27,570 You can work out a few things about it, but I'm just going to use it here to tell you a little bit about primes. 389 00:40:29,240 --> 00:40:36,050 It's got these nice properties. It basically takes multiplication and turns it into additions. 390 00:40:36,980 --> 00:40:43,100 So if you take a line of a product, you just add the values of that line. 391 00:40:44,090 --> 00:40:49,970 And it's not a line. Numbers aren't very big compared with what originally you had. 392 00:40:50,180 --> 00:40:53,530 So why a line of a million is just 13.8. Okay. 393 00:40:53,570 --> 00:40:59,870 So it takes quite large numbers and gives you quite small ones, but you don't really need to know much about it. 394 00:40:59,870 --> 00:41:10,580 But it does come up in this next theorem. So this is gives you a sense that actually primes do follow some sort of pattern. 395 00:41:11,600 --> 00:41:15,559 You might think to yourself, how many numbers are there? How many prime numbers are there? 396 00:41:15,560 --> 00:41:22,460 Less than a million or a billion or a trillion. And you can get pretty good estimates for that without actually working them all out. 397 00:41:23,390 --> 00:41:30,200 And that's what mathematicians suspected since 18th century proved at the end of the 19th century, 398 00:41:30,830 --> 00:41:37,820 that if you have want to work out how many prime numbers there are less than, say, about a billion. 399 00:41:38,570 --> 00:41:42,170 You work out a billion divided by a line of a billion. 400 00:41:43,010 --> 00:41:49,700 And that would give you a pretty good estimate of how many prime numbers there are without having to work them all out. 401 00:41:50,570 --> 00:41:53,620 And in fact, here's a picture of the graph. Okay. 402 00:41:53,620 --> 00:41:57,050 So I'm not sure you can actually see this in this slide, but there are actually two lines there. 403 00:41:57,920 --> 00:42:09,710 So the top line, the blue one, is how many prime numbers there are less than all these numbers and the N divided by. 404 00:42:09,890 --> 00:42:13,160 So X divided by a line of X is the estimate. 405 00:42:13,670 --> 00:42:16,640 And it's always a bit of an underestimate, at least initially. Okay. 406 00:42:16,970 --> 00:42:23,600 So it gives you a bit of an underestimate here, but it makes a pretty good estimate of how many they are. 407 00:42:23,900 --> 00:42:28,340 And long term, they're basically pretty much the same number. 408 00:42:28,390 --> 00:42:39,440 Well, what this twiddle is means on the previous slide is that if you divide how many prime numbers there are, 409 00:42:39,650 --> 00:42:43,520 by your estimate, long term, that number goes to about one. 410 00:42:44,120 --> 00:42:48,260 So pretty much is 100% long term, but it's estimated. 411 00:42:51,600 --> 00:43:00,120 Anyway, I'm almost done here. So let me let me try and give you a sense of, first of all, how hard it is to factor ice things. 412 00:43:01,990 --> 00:43:11,740 Using this prime number theory. From what I said to you, how many how many prime numbers am I going to have to try? 413 00:43:11,770 --> 00:43:21,370 How many factorisation I'm going to have to try to actually work out, say whether something in the order of ten to the 20 is actually prime. 414 00:43:21,430 --> 00:43:27,430 So I've got a 20 digit number, not a thousand digit number like we had in our say, a 20 digit number. 415 00:43:28,710 --> 00:43:36,690 To test whether it's really prime or not. I've got to try potentially all the prime numbers up to its square root. 416 00:43:37,710 --> 00:43:38,060 All right. 417 00:43:38,070 --> 00:43:47,460 And if you use the previous theorem to work out how many calculations I'd have to do, it ends up being about two RU ten divided by this number. 418 00:43:47,830 --> 00:43:56,280 And so just to give you an example here, if I was trying to work out whether a number with 20 digits actually was prime, 419 00:43:56,670 --> 00:44:03,300 and let's say it was I'm going to try all these that many calculations to actually work out where it was. 420 00:44:03,750 --> 00:44:08,100 I'm just for this 20 digit number. I've got to do 400 million. 421 00:44:08,370 --> 00:44:11,909 Trial factorisation of quite large numbers. Okay. 422 00:44:11,910 --> 00:44:15,600 So just a 20 digit number is going to be quite hard to check. 423 00:44:15,600 --> 00:44:22,110 It's prime. I might have to do 400 million calculations to actually work that out. 424 00:44:23,610 --> 00:44:29,220 There are over all sorts of ways of trying to test this, but it's always going to be a hard thing to check. 425 00:44:31,920 --> 00:44:37,860 So as I said here, there's actually it's actually quite easy to make claims about prime numbers that might be very hard to prove. 426 00:44:39,930 --> 00:44:52,790 Here are a few things we do know to be true. If you take all the prime numbers on and they are inverses, so I'll call a reciprocal. 427 00:44:53,760 --> 00:45:00,419 You take all the sums. That's some keeps getting bigger and bigger and bigger and basically goes off to infinity. 428 00:45:00,420 --> 00:45:10,140 So that becomes large. And Euler prove that if you take any what's called arithmetic progression. 429 00:45:10,920 --> 00:45:18,300 So something like perhaps three, seven, 11, 15, 19, 23 and so on. 430 00:45:19,260 --> 00:45:22,680 So you start with a certain number and then you have a common difference all the time. 431 00:45:23,940 --> 00:45:29,930 Now you can see here I've got various things that are prime. And I've got various things that aren't. 432 00:45:30,350 --> 00:45:36,209 Well, there's got to more primacy about the next one isn't. A man called Jerry Schley. 433 00:45:36,210 --> 00:45:41,490 He went and proved all these sequences will always have infinitely many primes in them. 434 00:45:41,820 --> 00:45:49,260 Okay, so for any sequence like that, yes, you'll get some things that aren't prime, but there will be infinitely many primes in those. 435 00:45:49,290 --> 00:45:49,940 He proved that. 436 00:45:50,700 --> 00:45:58,690 And a man called Chevy chef who was a Russian mathematician, he showed that there'll always be a prime number between any N and it's double. 437 00:45:58,710 --> 00:46:02,250 So if I gave you ten and 20, there's a prime number between those. 438 00:46:02,610 --> 00:46:06,450 If I gave you a hundred and $200, always be a prime number between those, perhaps lots of them. 439 00:46:07,290 --> 00:46:12,090 But you proved that. Always be a prime number before between those and then just to finish. 440 00:46:12,540 --> 00:46:16,050 Uh, these are things we don't know. 441 00:46:18,030 --> 00:46:23,820 Can't every hole number, every even number be written as the sum of two primes? 442 00:46:24,450 --> 00:46:30,630 If I gave you ten, you can write it as three and seven. If I gave you 12, that's five and seven. 443 00:46:31,870 --> 00:46:36,060 If I gave you 20, this is three and 17. 444 00:46:37,530 --> 00:46:42,210 Oh, you know, no one knows if every eve number can be written like that. 445 00:46:42,360 --> 00:46:46,290 We we've checked it up to about sort of like ten to the 20 or something like that. 446 00:46:46,290 --> 00:46:50,010 It's probably true, but only takes one counterexample to show it isn't. 447 00:46:50,240 --> 00:46:56,610 And no one knows the answer. No one knows whether it's possible to write every eve a number as the sum of two primes. 448 00:46:57,540 --> 00:47:02,190 Keep trying if you wish. But people, like I say, have gone up to ten to the 20, and that hasn't. 449 00:47:02,880 --> 00:47:09,240 I'm no one's found one yet. And then there are over important types of primes, primes of this form two to the P minus one, 450 00:47:09,660 --> 00:47:15,330 and no one knows whether they're infinitely many of those either. So problems are an important thing. 451 00:47:15,570 --> 00:47:20,490 The very easy to describe in some ways very useful in things like Internet security. 452 00:47:21,210 --> 00:47:26,190 But it's also very easy to ask very hard questions about them. So I think I'm going to finish there. 453 00:47:26,710 --> 00:47:30,030 Is anyone got any questions they might want asking? 454 00:47:30,510 --> 00:47:38,670 Like you say, if you want to find out more about how to create secure codes and how to scramble messages carefully, 455 00:47:38,970 --> 00:47:42,690 then Simon Singh's books, very good. If you want to have a read on this further. 456 00:47:43,170 --> 00:47:46,980 Okay. I think I'll hand over to Zarina if she's in the audience. Thank you.