Super Facts about 6-7 - Numberphile

0:00 So Brady, you must have seen uh out there on the internet

0:03 or in classrooms across the up and down the country the old uh 67.

0:07 So I'm going to tell you about that.

0:09 The meme that's sort of taken over the world and is

0:11 the bane of every school teacher across across the land.

0:15 I've heard of it, but what is it?

0:17 So yeah, I mean so I've got kids who are sort of, you know, like 12 and 14.

0:21 So so it's very sort of prominent there.

0:23 And it's Oh, double six AND SEVEN.

0:26 OH WOW.

0:26 I didn't even [laughter] notice that.

0:28 It means literally nothing.

0:30 It's like 67, so it's empty.

0:33 I mean, it has an origin, but but it's it's really So,

0:36 if you're if you're a teacher in a class and you dare say the number six,

0:39 the whole class is going to go 67.

0:41 And it's like, oh my god, it's so annoying.

0:44 So, there's like it's something you say and there's a motion.

0:47 You have to do a little motion.

0:48 Yeah.

0:48 Yeah.

0:48 I'm sure I'm doing it wrong, by the way, cuz I'm like some old old fella.

0:51 You've got kids in every classroom up and down the line doing six,

0:55 seven, six, seven, driving their teachers absolutely mad.

0:58 My my sister-in-law, she's a school teacher, uh, Cathy,

1:01 and she it's the thing that she hates most in the universe.

1:05 It's she absolutely hates it.

1:06 K Star got into trouble recently because he was

1:09 in a a classroom and uh I think they went to page six

1:13 in a book and he went 67 and uh it' been

1:16 banned in that school because the teachers were so annoyed by it.

1:19 So he got into trouble for that.

1:20 So it's up and down the land.

1:21 It's driving everybody mad.

1:22 But I think it's time to sort of reclaim

1:26 67 because of course they're numbers and numbers are math.

1:29 So let's do some interesting maths with six and SEVEN.

1:31 EH, 67.

1:34 OKAY.

1:34 So let's where should we start?

1:35 So 67 is a perfect prime pair.

1:40 So what do I mean by that?

1:42 So so that's so of course six is a perfect number.

1:45 So a perfect number is a number that's equal to the the sum of its divisor.

1:50 So let's take let's look at six.

1:52 Its divises are one, two, and three.

1:56 We won't and six, but you don't count the number itself.

1:59 You add these together, you get six.

2:01 So that's a perfect number.

2:03 28 is also a perfect number.

2:05 Divises of 28 are going to be 1 2 4 7 and 14, right?

2:12 And you add those together and you get 28 as well.

2:16 Perfect numbers are a big deal.

2:17 They're pretty rare.

2:17 And so obviously perfect prime pairs are rare because there six is is perfect.

2:23 Seven is prime.

2:24 Actually um 28 and 29 are a perfect prime pair.

2:28 Actually that's that's another one.

2:29 There aren't that many there.

2:30 There there I actually only could find a small

2:32 handful and I'm not clear if there's any more unknown.

2:36 This number 33,55036 that number is perfect.

2:43 And of course this number is prime.

2:46 Okay.

2:47 Okay.

2:47 So, that's the next one I'm aware of.

2:48 Then you've got this one as well, which is another another combination.

2:52 This number is perfect.

2:53 And this number's prime.

2:54 These are the only ones I know about.

2:55 So, he's got 67.

2:57 67.

2:57 There we go.

2:59 Yeah.

2:58 Got to do you got to do it right.

3:00 [laughter] Yeah.

3:01 67 28 29.

3:03 This one and this one.

3:04 And that's all the ones that I could find.

3:05 Perfect numbers are very rare actually.

3:07 They're not common.

3:08 They're not easy to find.

3:09 But you cleared smart guy that that he was came up with a way to find them.

3:13 So, you take a meen prime.

3:15 Okay.

3:16 So it's 2 to the p minus one.

3:18 So it's a prime of that form.

3:19 Okay.

3:19 So let's assume this is a mer prime.

3:21 And you can construct this number 2 p minus one.

3:24 Okay.

3:25 So this number you know all its uh all its factors.

3:29 So they are the mer prime itself.

3:31 This one here and then all the multiples of two that you get

3:35 from here right and you add them all together and you will get this.

3:37 This is what Uklid showed.

3:39 And this is this is a perfect number.

3:41 So this is this is a way to get not for all pays.

3:43 No, they have to be that has to be mer, right?

3:46 That has to be a mer prime.

3:48 Yeah, but which only happens for some ps which only happens for some PS.

3:52 And then what you could do then is is just look

3:54 through all these for all the mer primes that you know.

3:57 You can generate the corresponding perfect number

3:59 and then you can try and figure

4:01 out if adding one to it is going to give you a prime number.

4:03 So it's not not straightforward, but that's the search method you could do.

4:06 All of these ones by the way, they all fall into this category.

4:09 So the P's, this one's P equals 2.

4:12 You can check that.

4:12 So you get P to the 2- 1 is is 4- 1 is 3 and that's just going to give you a 2.

4:17 2* 3 is 6.

4:18 Okay, this one is P= 3.

4:20 This one is P= 13 and this one is P= 19.

4:24 These are all constructed this way.

4:26 So you notice all the perfect numbers in this in these examples

4:29 at least and actually these ones they're all even, right?

4:32 We don't know of any odd ones.

4:35 We don't know whether or not they exist.

4:37 We we think they pro possibly do,

4:39 but it's one of the mysteries of of mathematics

4:42 is whether there's an odd perfect number.

4:44 If it exists, it's got to be really really big.

4:48 Um it's got to be bigger than 10^ the 1500.

4:51 So if if it's if it exists, it's bigger than that.

4:54 If it does exist, it won't be part of a perfect prime pair.

4:58 Can't possibly be.

4:59 No, because it'll have an even next to it.

5:01 Yeah.

5:01 Absolutely.

5:01 Absolutely.

5:04 Okay.

5:03 So Okay.

5:03 So that's first.

5:04 Okay.

5:04 So we we won't stop there.

5:06 All right.

5:06 All right.

5:06 You want more paper for your next 67?

5:08 [music] All right.

5:13 So, 67.

5:15 Uh, [laughter] where else?

5:16 You can do the hands.

5:17 I got to do the hand.

5:18 67.

5:19 Where else does it appear?

5:19 Does it appear in pi, do you reckon?

5:22 Six and seven.

5:23 Of course it does.

5:23 Of course it does.

5:24 You know where?

5:26 Uh, does it appear in the first 100 digits?

5:31 Just It's the 99th and the hundth digits of pi.

5:35 Yeah.

5:35 So it's at it's at the 98th decimal place, but it's the 99th digit.

5:39 It's the six.

5:40 And then the seven is is at the hundth one.

5:42 So there you go.

5:44 It's also in E, of course.

5:45 Um it's the it's the 60th and the 61st digit of E.

5:50 Probably elsewhere in E.

5:52 Yeah, definitely.

5:52 Of course, it's it's it's elsewhere in pi as well.

5:54 It's the it's the 235th and the 236th digit of pi.

5:58 Again, it obviously crops up again.

6:00 I did wonder if it was the sixth and seventh um

6:04 digit of any number and it is uh something called champion's constant.

6:10 Uh but it's a bit of a cheat.

6:12 Um heard of that.

6:14 Yeah, it's a bit of a cheat.

6:15 You'll be annoyed by this because

6:16 of course Champan's constant you construct like this.

6:18 It's 0.1 2 3 4 5 6 7 8 9 10 11 and you construct it like that.

6:28 Right?

6:28 So it's almost by construction that it's this they discount the zero.

6:31 It's the sixth and seventh.

6:33 You don't like that one?

6:34 No.

6:35 Well, it it is a constant which has six six seven as a six seventh position,

6:39 but it's the only one I could find.

6:41 Well, there be an infinite number of numbers where six and seven.

6:43 Yeah, but are they interesting numbers?

6:44 Are they are they are they named?

6:45 Are they significant in any way?

6:47 That's the question, right?

6:48 So, um 67, we should also talk about 67 is obviously another interesting number.

6:54 Now, 67 is a prime number.

6:58 Okay.

6:58 It's also a sexy prime number.

7:00 So sexy prime is a prime number that is six places away from another prime.

7:06 But more than that, it's part of a sexy prime.

7:09 Triple Brady.

7:11 Triple sexy.

7:11 Triple sexy.

7:12 Yeah.

7:13 Show me.

7:13 Say I'll show you.

7:14 So well, let's just say and it's actually sat right in the middle of it.

7:17 So 61.

7:18 It's in a sexy prime sandwich.

7:19 It is in a sexy prime sandwich.

7:21 So 61, 67, and 73.

7:24 These are all sexy primes differing by six.

7:27 And and if you can see our our 67 sitting there right in the middle there.

7:30 So it's a it's a sexy prime and it's a you

7:33 know right in the middle of it little sexy pine triple.

7:35 So there you go.

7:36 Um what else?

7:37 It's not just a prime really.

7:39 It's it's not just a sexy prime.

7:41 It's a super prime as well.

7:43 So what's a super prime?

7:45 So if you write out all the prime numbers and then uh so

7:48 obviously 67 we know it's a prime number so it's going to be there.

7:50 And we ask where are you in the prime number table?

7:53 Okay, so 67 is the 19th prime number and 19 is a prime number.

8:01 So that makes 67 a super prime.

8:04 Okay.

8:05 So you know what I'm going to ask?

8:09 What?

8:08 What's the 67th prime number?

8:11 Should we check?

8:12 [laughter] Yeah, it's it's 331.

8:14 Really?

8:16 Okay.

8:16 It's disappointing that.

8:17 No, because you Oh, six.

8:18 Yes.

8:18 Yes, I see it.

8:19 Yeah.

8:19 You add [laughter] the two threes, you get a six.

8:21 Yeah.

8:21 And the other one, you get a seven.

8:22 Yeah.

8:23 Yeah.

8:23 Yeah.

8:23 Yeah.

8:23 Yeah.

8:23 Yeah.

8:23 Yeah.

8:24 Okay.

8:24 Cool.

8:24 Cool.

8:24 We got it.

8:25 We got it.

8:25 [laughter] Yeah.

8:26 All right.

8:27 All right.

8:27 Excellent.

8:27 So, we've got super primes.

8:28 67's a super prime.

8:30 It's not a super super prime then.

8:32 So, a super super prime is a prime number at a super prime position.

8:37 So, they do exist.

8:38 H there are plenty of of super super primes.

8:40 You of course carry on like this.

8:42 You could say super super super prime which is obviously

8:44 a prime number at a super super prime position and so on.

8:47 You could you could really sort of imagine this.

8:49 So, so what's true is that if I take super I don't know to the K prime,

8:56 so I've got I've got K supers, right?

9:00 And K is some finite number,

9:01 then this will always there'll always be

9:03 an infinite number of numbers that will satisfy this.

9:06 But if I take K to infinity,

9:08 I'm going to conjecture that there are there are no numbers that satisfy it.

9:13 I think that's true.

9:14 It's kind of weird, right?

9:15 So you've got an infinite.

9:16 So for any finite K, I'll have an infinite number of super to the K prime.

9:21 Super super any finite number.

9:23 Doesn't matter.

9:24 I can take three three of these guys.

9:25 Yeah.

9:26 And there'll still be an infinite number of them.

9:28 But if I take the number of supers to be itself infinite, I end up with nothing.

9:34 Nothing survives.

9:36 It's kind of weird, isn't it?

9:37 The lowest number in the table is going to just

9:40 grow and grow and grow and grow and grow with K.

9:42 And so until you take K to infinity,

9:44 then it just it just falls off the page essentially.

9:46 You see what I mean?

9:47 I'll let the set theorist

9:48 let the set theorist figure it out, but that's my conjecture.

9:51 All right.

9:52 Okay.

9:52 The Padilla conjecture.

9:54 Is it Can we call it Pila conjecture?

9:56 Yeah, let's call it conjecture.

9:58 If I don't know if it's been conjectured before.

10:00 I don't know if it has.

10:01 It could be yours.

10:02 Yeah, let's have it.

10:03 [laughter] We can share it with you.

10:04 Really?

10:05 All right.

10:05 It's Christmas, right?

10:06 Let's share the love.

10:07 All right.

10:08 Okay.

10:08 67 67.

10:09 67.

10:10 Uh, so 67.

10:11 Another cool thing about 67.

10:12 Now, my wife told me not to tell you about this one,

10:14 but I like it and I think it's good and I think you'll agree Brady.

10:18 So, let's let me tell you this one, right?

10:23 You have to take you have to construct

10:24 the prime orals which are products of primes.

10:28 Okay?

10:29 So, so the nth primeal is just 2* 3* 5 until and then

10:35 you carry on taking these products until you get to the nth prime.

10:39 Okay?

10:39 Okay, so the fortunate numbers are what you get.