I posted something on Twitter a while ago

about a really crazy number

with a lot of digits that are all the same

and it got a really huge response.

I just want to share with you,

my experience with this number.

Let me draw the number for you

Here we go

Here we go

Here we go

And in the words of Andrew Wiles

"I think I'll stop there"

This number here

All nines. You can see it. In actual fact,

the way that you'd describe this is as a repdigit.

So, in the maths canon you'd say repdigit, meaning "repeated digit".

But what's special about this

There's two things really special about this number

is that here

down here, there's one digit that's changed

BRADY HARAN: So, it's not a repdigit?

SIMON: It's called a near repdigit

So, it's just really close

But you know what I like to call it, Brady?

I like to call it a glitch.

I feel like it's a glitch,

because it's not only a glitch number.

Believe it or not, this is a prime number.

This here, I posted this on Twitter

It really got people going

You know what I thought, Brady?

I thought

Am I sure I haven't made a mistake?

I'd really like to check this number,

you know?

As a mathematician, that's what you wanna do, you want to check it.

The only problem is: how do you check a number like this?

I- I really don't suggest you stick this into the Google entry line.

To actually work out if it was a prime or not,

I actually needed to "compactify" it

I mean, I needed to actually go

I needed to actually transform it into something that I could carry around

So, this number has 506 digits

The 8 here, if you count from the bottom

From the ones column, and the tens column, and the hundreds column

If you get up, you actually get to the 254th column

So, it's actually 254 along

So, it's just over halfway

How can we actually transport this in a way that doesn't involve me writing out all these nines?

Any ideas?

What's really cool about it is the fact that they are nines.

So, if you add 1 to 9,

you get a ten.

And, if you add 1 to 99,

you get a hundred.

And, if you add 1 to 999, you get a thousand.

So, we can actually use that fact,

because, if we can actually get this as

something to the power of 10,

well, that can really make things a lot easier for us to handle.

So, if we think about it, right

this has 506 nines.

So, if we say 10

take 1 is 9,

10 squared take 1 -- which is 100

take 1 -- is 99.

And, 10 cubed which is,

three zeros there, take 1, is 999.

Then, if we go, dot, dot, dot...

-- You gotta love the dot-dot-dots, because you can put anything in dot-dot-dots --

we can then say

10 to the 506 take 1

is equal to... -let's just jump to the chase, hey, Brady?-

... is equal to nine-nine-nine, dot-dot-dot,

nine-nine-nine, where this

happens to be

506 digits.

So, that's kind of that, except for that

BRADY: I know that's a small change, but that actually changes the value of the number quite a lot, doesn't it?

SIMON: What do you mean, "changes the value"?

BRADY: The quantity, like, if you were going to buy that many eggs...

SIMON: [laughs] You wanna put this into eggs?

BRADY: ...changing that from an 8 --

Changing that from a 9 to an 8 will dramatically change the number of eggs you just bought.

SIMON: Well, you know what? You're exactly right,

and to actually work out by how much,

you're actually solving the problem.

9 take 8... is 1.

90 take 80, that's 10.

900 take 800 is 100.

Can you see the pattern going on here?

So, if we actually think there's a 9 right there,

in the 254th position...

So what would that mean? That means that we're actually taking away 1,

with this many zeros.

So, we gotta imagine there's a 1 here,

for every 9 there's basically a 0, because that's continuing what's going on here.

So now what we're doing is we go 10

to the 506 take one

gives us this sea of 9s, then we just want to take 1

with a bunch of zeroes so we get it up to that point there. So we're basically taking away

10 to the 253.

So that's what that's equal to.

So it's that take 1 to give us the 9s take that

To take one away from that spot.

So yeah, I found a way to actually carry it around and talk to it. This is the way you talk about big numbers, big prime numbers to your friends.

You don't go 99999999

Some sort of Ali G reference

99999999999

which is interesting. In actually fact, Ali G does this thing where he goes

"What's 9999999 take 9999989999" and he actually does this and I actually counted the number times he said 9s and 8s

8999999

and it's unbelievable, he's like 1 off

1 place off from actually having one of these glitch primes.

That would have really made the nerds excited.

So now we have a way to actually understand this number. Now we have

10 to the 506

take, and I'm just going to order it in terms of magnitude here, take that,

Basically what you need to notice here, what's going on,

is the fact that if you double 253 guess what you get?

Two threes are six, two twenty-fives are - it's actually that number

So now we can actually find a structure. I'm going to put in a couple letters here to denote other things. I can make it even smaller if you check this out.

Maybe not, because in actual fact for m

don't you love mathematicians

This just makes it more complicated.

So in actual fact, it actually has this shape. So what happened next?

I started doing this at 9PM, I was still doing this at 6AM.

I mean this is the hole I want to take you down.

So now, what's really interesting is that if you learn about how to factorize polynomials or even factorize things because this is actually 10 m squared.

take 10 m take 1

so you kind of get this taught at school. It actually has the shape of x squared take x take 1.

So you actually think maybe I'm going to be able to work out a way to see when this is a prime or not

See this is what's amazing about these primes.Primes don't act that way.

In actual fact, I needed to put this number into a database that would crunch it for me

and you wouldn't believe

there's only a small number of m's.

which will actually give you this particularly shaped glitch prime.

And they happen to be, for values of m

where m is equal to 1

6

9

154

253 hello

1114

and 1390

and that's as far as the database went. And I think that's probably as far as we know at this point

for this particular-. So look at that! Think about how big this number is, how many values of m you can have and it's only for these numbers

that this will be prime, and that happens to be prime

What's cooler than just a glitch? No glitch! Well

Maybe

But maybe what happens if the glitch was right in the middle of the number?

So I actully looked into that as well. That's a different formula.

So that formula then is, we're saying is 10 to the 2m

plus 1 take 10 m

take 1. It basically has to be an odd number of digits.

Not an odd number, but in terms of the number of digits it has to be an odd number because it has to have a middle.

So there are lots of different values for that but there's a really nice value. The first one that we can find is actually

m equals nine

so what it looks like is one, two, three, four, five, six, seven, eight

and nine

eight

and then

one, two, three, four, five, six, seven, eight, nine

So it's nine 9s, 8, nine 9s.

Come on. Swish!

See, I looked at this and I thought, "Glitch"

What are glitches, when we think glitches we think computers, we think binary.

So really, where this took me to, is like I'm looking at glitch primes the wrong way, I should be looking at binary glitch primes.

So I want to look at binary numbers. So in that fact, in that case, in looking at instead of 9s, I'm looking at one one one one... glitch... one one one one

and also let's just think about the coolest thing: a palindrome.

And they've actually got a special name. They're called Cyclops numbers.

So for example: 5 is 101.

Cyclops because it's got a hole in the middle.

But you've got lots of other Cyclops numbers. I mean I don't even have to work it out you can have like

two 1s and a 0 and two 1s.

Whatever that is.

Or you could have three 1s and a 0, three 1s.

Or you could have like 4, and a 0 and 4.

I mean this is getting-, I mean this is the sort of stuff you communicate with aliens about, you know what I mean?

5 is a prime. Are those other ones prime?

Well that was my question.

I wanted to know if other ones could be prime.

And this is really where my mind was blown.

Because I then had to change the structure of this thing here, this 10

because it was base 10, but if we actually do it like this

Because then if it's 2m, because we're basically doing it in binary so it means we need to do it in power 2.

Right. What that means is, this is unbelievable, I'm able to actually factorize this

into 2 m plus 1

plus 1

to 2

m take 1

So you're always able to do that because it's two. You can't do that with any other number higher than two.

And this is a problem because, guess what?

Primes, by definition,

don't have any factors.

They don't have any factors. They'll only divisible by themselves and one.

So if you actually turn this always into this

the only time this is not going to be a prime is when one of these things is equal to one.

So this is getting added, it has to be 2m take 1 equals 1.

So, that means it's 2m equals 2, oh come on, it's m equals 1.

So that means it's uh, that means the only prime we have is

2 to the power 2 plus 1 -- 5.

So, where this took me. Unbelievable. Morning. The sun's rising. And I realized, maybe I'm the first person to realize (not really but that's how I feel about maths)

that the only glitch prime you can have in binary is 101.

We'd like thank Audible.com for their support of this Numberphile episode.

If you know or enjoy a book or spoken material, Audible has everything you could wish for

They've got over 180,000 titles to choose from

and they're offering a 30 day free trial

when you sign up for the service

Now if you're in the mood for some more mathematics I can high recommend these books on the screen, including Alice's adventures in Numberland and the Mathematics of Love

both by people who appear in Numberphile videos.

Or if all this talk of Cyclops has you excited why not go old school and listen to none other than the Odyssey by Homer

Not that homer.

And if you do check out Audible, and I highly recommend I use Audible myself.

Then use the URL audible.com/numberphile

That means if you do take them up on that free trile, the good people there will know you came from here.

That's audible.com/numberphile and our sincere thanks to them for helping us make more numberphile videos.