Super Facts about 6-7 - Numberphile

[00:02] across the up and down the country the old uh 67. So I'm going to tell you about that. The meme that's sort of taken over the world and is the bane of land. >> I've heard of it, but what is it?

[00:16] >> So yeah, I mean so I've got kids who are sort of, you know, like 12 and 14. So so it's >> Oh, double six AND SEVEN. >> OH WOW. I didn't even [laughter] notice that. It means literally nothing. It's

[00:30] that. It means literally nothing. It's like 67, so it's empty. I mean, it has if you're if you're a teacher in a class and you dare say the number six, the whole class is going to go 67. And it's like, oh my god, it's so annoying.

[00:44] and there's a motion. >> You have to do a little motion. Yeah. the way, cuz I'm like some old old fella. You've got kids in every classroom up and down the line doing six, seven, six, seven, driving their

[00:57] sister-in-law, she's a school teacher, uh, Cathy, and she it's the thing that she hates most in the universe. It's she absolutely hates it. K Star got into trouble recently because he was in a a classroom and uh I think they went to

[01:12] page six in a book and he went 67 and uh it' been banned in that school because he got into trouble for that. So it's up and down the land. It's driving and down the land. It's driving everybody mad. But I think it's time to

[01:25] sort of reclaim 67 because of course they're numbers and numbers are math. So and SEVEN. EH, >> 67. 67 is a perfect prime pair.

[01:40] So what do I mean by that? So so that's so of course six is a perfect number. So a perfect number is a number that's equal to the the sum of its divisor. So let's take let's look at six. Its divises are one, two, and three. We

[01:56] won't and six, but you don't count the number itself. You add these together, you get six. So that's a perfect number. 28 is also a perfect number. Divises of 28 is also a perfect number. Divises of 28 are going to be 1 2 4 7 and 14,

[02:12] right? And you add those together and you get 28 as well. Perfect numbers are a big deal. They're pretty rare. And so obviously perfect prime pairs are rare because there six is is perfect. Seven is prime. Actually um 28 and 29 are a

[02:27] that's another one. There aren't that many there. There there I actually only could find a small handful and I'm not clear if there's any more unknown. This clear if there's any more unknown. This number 33,55036

[02:44] this number is prime. Okay. Okay. So, that's the next one I'm aware of. Then another another combination. This number is perfect. And this number's prime. >> 67. There we go.

[02:58] [laughter] >> Yeah. 67 28 29. This one and this one. find. Perfect numbers are very rare actually. They're not common. They're not easy to find. But you cleared smart guy that that he was came up with a way

[03:12] to find them. So, you take a meen prime. Okay. So it's 2 to the p minus one. So let's assume this is a mer prime. And you can construct this number 2 p minus

[03:24] one. Okay. So this number you know all its uh all its factors. So they are the its uh all its factors. So they are the mer prime itself. This one here and then from here right and you add them all together and you will get this. This is

[03:38] what Uklid showed. And this is this is a perfect number. So this is this is a way that has to be mer, >> right? >> but >> which only happens for some ps

[03:50] >> which only happens for some PS. And then what you could do then is is just look through all these for all the mer primes that you know. You can generate the you can try and figure out if adding one

[04:02] number. So it's not not straightforward, but that's the search method you could do. All of these ones by the way, they all fall into this category. So the P's, this one's P equals 2. You can check that. So you get P to the 2 - 1 is is 4

[04:15] - 1 is 3 and that's just going to give you a 2. 2 * 3 is 6. Okay, this one is P you a 2. 2 * 3 is 6. Okay, this one is P = 3. This one is P = 13 and this one is P = 19. These are all constructed this way. So you notice all the perfect

[04:27] numbers in this in these examples at least and actually these ones they're all even, right? We don't know of any odd ones. We don't know whether or not they exist. We we think they pro possibly do, but it's one of the

[04:40] mysteries of of mathematics is whether there's an odd perfect number. If it exists, it's got to be really really big. Um it's got to be bigger than 10^ the 1500. So if if it's if it exists, it's bigger

[04:53] than that. >> If it does exist, it won't be part of a >> because it'll have an even next to it. >> Yeah. Absolutely. Absolutely. >> So Okay. So that's first. Okay. So we we won't stop there.

[05:06] >> All right. All right. You want more paper for your next 67? [music] >> I got to do the hand. 67. Where else does it appear? Does it appear in pi, do

[05:20] >> Six and seven. Of course it does. >> Of course it does. You know where? >> Of course it does. You know where? >> Uh, does it appear in the first 100 digits? >> Just It's the 99th and the hundth digits

[05:33] >> Yeah. So it's at it's at the 98th decimal place, but it's the 99th digit. It's the six. And then the seven is is at the hundth one. So there you go. It's also in E, of course. Um it's the it's the 60th and the 61st digit of E.

[05:50] >> Yeah, definitely. Of course, it's it's it's elsewhere in pi as well. It's the it's the 235th and the 236th digit of pi. Again, it obviously crops up again. I did wonder if it was the sixth and seventh um digit of any number and it is

[06:07] uh something called champion's constant. Uh but it's a bit of a cheat. Um heard >> Yeah, it's a bit of a cheat. You'll be annoyed by this because of course Champan's constant you construct like this. It's 0.1 2 3 4 5

[06:24] this. It's 0.1 2 3 4 5 6 7 8 9 10 11 and you construct it like construction that it's this they discount the zero. It's the sixth and >> You don't like that one? No. >> Well, it it is a constant which has six

[06:37] six seven as a six seventh position, but it's the only one I could find. >> Yeah, but are they interesting numbers? Are they are they are they named? Are they significant in any way? That's the question, right? So, um 67, we should

[06:50] also talk about 67 is obviously another interesting number. Now, 67 is a prime interesting number. Now, 67 is a prime number. Okay. It's also a sexy prime number. So sexy prime is a prime number that is six places away from another

[07:05] prime. But more than that, it's part of a sexy prime. Triple Brady. >> Triple sexy. Yeah. Show me. Say I'll it's actually sat right in the middle of it. So 61.

[07:18] >> It's in a sexy prime sandwich. >> It is in a sexy prime sandwich. So 61, 67, and 73. These are all sexy primes differing by six. And and if you can see our our 67 sitting there right in the middle there. So it's a it's a sexy

[07:32] middle of it little sexy pine triple. So there you go. Um what else? It's not just a prime really. It's it's not just a sexy prime. It's a super prime as a sexy prime. It's a super prime as well. So what's a super prime? So if you

[07:46] uh so obviously 67 we know it's a prime number so it's going to be there. And we ask where are you in the prime number table? Okay, so 67 is the 19th prime

[07:58] table? Okay, so 67 is the 19th prime number and 19 is a prime number. So that number and 19 is a prime number. So that makes 67 a super prime. Okay. So >> What? >> What's the 67th prime number?

[08:11] >> Yeah, >> it's it's 331. Really? >> No, because you >> Oh, six. Yes. Yes, I see it. Yeah. >> Yeah. And the other one, you get a seven. Yeah. Yeah. Yeah. Yeah. Yeah.

[08:23] We got it. [laughter] Yeah. All right. >> All right. Excellent. So, we've got super primes. 67's a super prime. It's not a super super prime then. So, a super super prime is a prime number at a super prime position. So, they do exist.

[08:38] primes. You of course carry on like this. You could say super super super prime which is obviously a prime number at a super super prime position and so on. You could you could really sort of imagine this. So, so what's true is that

[08:51] if I take super I don't know to the K prime, so I've got I've got K supers, right? And K is some finite number, then this will always there'll always be an infinite number of numbers that will satisfy this.

[09:06] >> But if I take K to infinity, I'm going to conjecture that there are there are no numbers that satisfy it. I think that's true. It's kind of weird, right? So you've got an infinite. So for any finite K, I'll have an infinite number

[09:19] of super to the K prime. >> Super super any finite number. Doesn't >> And there'll still be an infinite number of them. >> But if I take the number of supers to be itself infinite, I end up with nothing.

[09:34] Nothing survives. >> It's kind of weird, isn't it? The lowest grow and grow and grow and grow and grow with K. And so until you take K to infinity, then it just it just falls off the page essentially. You see what I

[09:47] >> let the set theorist figure it out, but that's my conjecture. >> The Padilla conjecture. >> Is it Can we call it Pila conjecture? >> If I don't know if it's been conjectured before.

[10:00] >> Yeah, let's have it. [laughter] >> We can share it with you. Really? All >> Okay. >> 67 67. >> 67. Uh, so 67. Another cool thing about 67. Now, my wife told me not to tell you

[10:13] about this one, but I like it and I think it's good and I think you'll agree Brady. So, let's let me tell you this one, right? You have to take you have to construct the prime orals which are products of

[10:27] the prime orals which are products of primes. Okay? So, so the nth primeal is primes. Okay? So, so the nth primeal is just 2 * 3 * 5 until and then you carry to the nth prime. Okay? Okay, so the fortunate numbers are what you get.