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Number Theory Primes · Acedra
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Number Theory Primes
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0:00
So you're sitting there all by yourself and at the next table you notice there's Freeman Dyson, there's Armand Borel; and they're having their own lunch very nicely and quietly but then they start arguing, and Borel says look give me a prime. Dyson says 2 to the 31 minus 1. Armand says no, I want the actual digits, don't don't give me any exponents, I want to actually write down the prime. So Freeman Dyson says, all right, here's a prime 111 111 111 111 111 1,
0:37
19 times, nineteen 1s. There's a prime for you. And Borel says, it's getting there but I need at least 20 digits, and they can't be all 1s. And they think for a minute and there's some a bit of silence and you cough quietly and you say, gentlemen I think I can help you, here's a prime for you. Twenty digits: 1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1.
1:11
That decimal number is a prime gentlemen. And Borel closes his eyes and thinks for a minute, says could be? Looks like you're right. And Freeman Dyson says, well congratulations young man, do you know any other primes like that? And you say well you can look general- more generally at what you get if you go out up to n. So let's write down the decimal number 1 2 3 4 5 6 7 8 9 10 10 11 12, go all the way up to n, and we're going to string all these digits together and then we go down n minus 1,
1:54
n minus 2, down to 4 3 2 1. So that's a number, rather large number if n is large, and it depends on n. - (Brady: Can I ask a quick question?) (Obviously) (11 counts as 1, 1 and 12 counts as) (1,2 and when we go back down the) - Exactly (slide will 12 become 2, 1?) No. 12 becomes- it stays at 1, 2. So so we read the numbers just as they are on the blackboard and they go up and then they go down. So so it goes- the the last would be 11 10 9 8 7 6 5 4 3 2 1 stop. And so
2:35
you can say when is that a prime? (For what n?) - For what n is that a prime? Well 10 works, n equals 10 gives a prime. And Borel says, and what's the next prime? You say you have to go out to n equals 2446 to get another prime. And beyond that we don't know. So this is a pretty large prime and we don't know anymore. So Brady, it's a story you can tell at parties.
3:16
And of course we would like another few terms so we could- the sequence is not yet- the sequence 10, 2446 is not yet in the OEIS because we only have two terms, it's too short. - (How many terms does) (it have to have before you put it in the) (Online Encyclopedia?) - There are some sequences where one term is enough, that are really really important. But but for something like this more than six digits if you can. I mean a couple more terms would be perfect. Officially five terms but we'll
3:50
certainly make exceptions if the terms are big. - (Neil that prime that has n) (equals 2446 must be a huge) (number) - It's a huge number. (Should we believe there are an infinite number of) (ns this will work for?) - Yes, yes absolutely. You can do the- do the math, you can look at the probability that a number like this is a prime and it's the probability is is one over whatever it is, and it dive- that summation diverges. So we expect with confidence that there'll be infinitely many. It gets
4:21
harder as the numbers get bigger and and presumably the man who found 2446 took it some distance further but I don't think anyone has seriously looked into this. I call this one a memorable prime, the one with 10 in the middle. I'm not finished, I have another segment. This is the most wanted prime in the world. So you saw the primes we just did, that you start at 1 and you go 1 2 up to n and then you go down n minus 1 down to 1. But surely the more logical question is, what
4:58
if you start at 1 and go up to n and stop? So let's do it: 1, that's not a prime. 12, that's not certainly not a prime; 123, well it's not a prime it's it's 3 times 41. 1234, obviously if it end- we're gonna go up to n and stop, and if n is even there's not a chance. If n is 5 there's no chance that it's a prime and if the sum of the digits is 3 there's no chance it's a prime. 6, that's most definitely not a prime. What about 1234567? That looks like it might be a
5:33
prime. Well it's not, it's divisible by 127, sorry. So we keep going, we want a prime, we definitely want a prime. This really is- you know, it's sort of the most natural prime for anyone with 10 fingers. 10 11; 11 doesn't work either. 12 13; so let me talk about 13 for a minute, so go 12345678 1- 10 11 12 13, these are just the digits of this number okay. And we say, all right now that really looks like it's a prime - it's not, it's divisible by 113, sorry. Well all right, so you keep going
6:18
up to n and every time you get to n you stop and think, and when n gets large you have to think quite a while before you can tell whether it's a prime or not. So various people help me with this and then I asked the people who run the GIMP, the Great Internet Mersenne Prime Search who've been looking for Mersenne primes, because this is exactly their kind of problem. And they took it up, a group of them worked on it for quite a while. And they got up to n equal to 340,000
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and they still hadn't found a prime. And I'm afraid they've given up; because do the math, look at the chance that a number like this up to n is a prime, it- roughly it looks like there's a 50/50 chance. By the time you get up to a million you should have hit a prime. So I would like to encourage the world to keep going and find find this missing prime. This episode has been brought to you by climeworks. We're all pretty conscious of the amount of carbon dioxide being put into the atmosphere,
7:28
climeworks is an innovative company doing something about it by literally capturing CO2 from the air itself. It's then reused by other businesses or permanently stored underground, basically as mineralised stone. Here's some of their kit in action, it's in Iceland, and you can be a part of it. They have various plans where you pay monthly to have carbon dioxide captured in your name. You will know exactly how much CO2 was removed from the atmosphere directly
7:59
because of you; or if not you this could be a great present to give that person in your life who doesn't want another pair of socks or endless boxes of chocolates, but they would like to know that they're part of a planet-saving mission. Anyway to find out more, go to gift.climeworks.com/numberphile The website's actually really good for showing how it all works. There's the link on the screen and of course there will be a link in the video description. Our thanks to climeworks.
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