We like doing videos about special numbers and we're gonna do a prime! Because we do love our prime numbers as well.

It's quite a big prime number. So I'm gonna write it out. So it's going to start:

357 sextillion

So it's already going to be big.

686 quintillion

312 quadrillion

646 trillion

216 billion

567 million

629 thousand

one-hundred and thirty-seven.

So it's a massive 24-digit prime number, but this is why it's special:

Because if I remove the first digit, the three, from that number

the remaining number here, starting with the five, is also a prime number.

And if I do that again, if I remove the five,

the remaining number starting with the seven is also a prime number.

And that will be true all the way down.

So up to here

629 thousand one-hundred thirty-seven is a prime number.

I can remove the digit.

The next number will be prime, all the way down to seven.

So it's always prime when I truncate from the left,

so it's called a left-truncatable prime.

And there are finitely many of them, and that is the largest one.

And I learned this from my friend Rob Eastaway, who has been on Numberphile

He's a mathematician as well, and he gave me this pencil with the number written on it

There is the number printed on the pencil. So no matter how much I sharpen this pencil

I'm always going to have a prime number, which I think it's why it's written on the other side of this pencil

There you go. "Always in our prime."

It's a kind of a joke.

Kind of a joke.

So this is a left-truncatable prime.

Let me show you how you can create them, 'cause they're not that hard to create.

So let's start with the last digit.

So the last digit could be,

You know. 1, 2, 3, 4, 5, 6, 7, 8 or 9.

My last digit could be any of these numbers, but it has to be prime as well

'cause I want a prime at the end.

So One is not a prime.

Two, we could have a prime Two at the end,

But that's not going to be able to be extended very much.

We could have Three as a prime number at the end, Four we're not gonna have at the end.

Five could be at the end. Six, no, not a prime.

Seven, maybe.

Eight, not a prime, and Nine is not a prime.

So it's going to have to end with one of these numbers, but let's just try now to extend it back.

So we're going to extend it back to the left and make a two-digit number

Let's go with the seven

Okay, let's make a chain here. Let's go from the seven

Well, then if I extend it, it's going to be either 17 or 27 or 37...

47 or 57 or 67 or 77...

or 87 or 97, so which of these are prime?

17 is a prime.

27, that's not prime.

37 is prime

47 is prime

57 is not

67 is prime.

77, that's not a prime.

87 is not a prime, 97 is a prime.

So let me just do a couple more steps,

let's go from the 47 and extend that further back.

147 perhaps, or 247...

347, 947... And we keep extending this if we want, let's find a prime on that list.

So, 947 is a prime. I'll go and extend that back another step.

So this could be 1947 perhaps, or 2947...

Or 9947.

I'm gonna extend another step here. I'm gonna go from this number,

I know this is a prime. So I'm going to go from here next,

try and extend that back, just do another step

So this is 3947, three thousand nine hundred and forty-seven

so that could now be 13947

or 23947

...93947.

And I'm going to stop at this step because if we do the checking through

None of these are prime

This 13947 is not a prime, 23947 is not a prime,

33947, not a prime

Not a prime.

None of these are primes. So the chain stopped. The chain is terminated.

So the last prime we had there in that chain was 3947.

So that's the end point for the chain that we had

The number of endpoints that you can make just doing this kind of method

There's 1442 of these endpoints.

So,

those are going to be the largest prime in the chain

And the largest one of them is this 24 digit number I've written out, but clearly you can see

there are only finitely many, because once you get to an end point, they can't be extended any further to the left.

So that's left-truncatable primes. Shall we do right-truncatable? Just to show you the largest one that's possible.

I'll show you the largest right-truncatable prime so this is just a bit of fun really so it's a silly thing

So this is just a bit of fun, really. It's a silly thing, I know.

However, if I used a different base, 'cause this is a base 10 thing,

I use a different base,

If I used a larger base - base twelve, or base 20 or base 100 or something like that

then each step here would have longer lists.

And if the list is longer, then they're more likely to hit a prime

So you're going to end up with longer chains, you actually end up with bigger

left-truncatable primes, if you do it in a different base or a larger base.

For right-truncatable primes, the biggest one we have is

73 million

939 thousand

one hundred and thirty-three.

So now if I start removing digits from the right-hand side

we will always have a prime. If I allowed One to be a prime,

we could have a larger number.

We could have this number:

One billion

979 million

339 thousand

three hundred and thirty-nine.

So this is a prime number, and if we remove digits from the right, we always have a prime number.

But we end up with One and One isn't a prime number. So I don't know why I even mentioned it.

Let's have a look at left- AND right-truncatable primes. Is there a prime that's on both lists?

And yes, there is.

That's the largest one of those that we can find:

739 thousand

three hundred and ninety-seven.

So that is left-truncatable and it's right-truncatable, so it's on both the lists.

Although we... I don't think we can truncate simultaneously

because we're not going to get prime numbers if we do that.

BRADY: What's the longest you can do simultaneously?

DR. GRIME: Well, I...

That is an interesting question, because what about if we can remove the digits in any order we want?

415 thousand six hundred and seventy-three.

And I'm gonna delete not just from the ends. I'm going to delete the One.

There, it's gone.

So I've got this number now: 45 thousand six hundred and seventy-three, and that's a prime number.

Now I'm going to delete another digit. I will delete the Three for this one.

So now I've got four thousand five hundred and sixty-seven

Now if I've got that, I'm gonna delete the Five in this case to get four hundred and sixty-seven, which is a prime.

And now I'll delete the Four

So to make 67, which is a prime. And now I think I'll delete the Six to get a Seven

So okay, I create this chain of primes, but I'm allowed to delete the digits any way I want,

so those are called deletable primes

and it is thought that there are infinitely many of those,

but that is something we don't know, 'cause we haven't proven that to be true.

So that is a challenge.

BRADY: What's the number you could create a deletable prime where it doesn't matter what digit you delete,

any digit you delete will still leave you with a prime number?

DR. GRIME: That's another that's another good idea.

I don't know if they exist.

That would be interesting to find out.

BRADY: Get to work

DR. GRIME: Oh, me?

BRADY: Now from past experience I can imagine some people might be thinking

"What's the point of all of this? Why study these truncatable primes and deletable primes?"

Well first I'm not entirely sure there has to be a point.

Sometimes things can just be fun

But I think there is a bit of a point too and let me explain

Studying problems like this, things from completely out of left field, make you think differently.

Mathematicians come up with new tools and ideas. And thinking differently,

Thinking about problems you don't normally think about is really good for your brain,

It makes you smarter. And this is where today's episode sponsor Brilliant come in.

Brilliant is a website full of quizzes and puzzles and courses, and they're all things that come out of left field.

They make you think differently. You can't rely on the equations or the principles you learned at school.

They challenge you and they make you smarter. Now, If you go to brilliant.org/numberphile, you can check it all out.

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