 ## 字幕表 動画を再生する

• We're going to do a silly one today, we're going to talk about a special number.

• 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.

• There's lots of stuff for free but if you use the /numberphile URL

• a) They'll know you came from here, and b) You can get 20% off their premium service which unlocks loads of extra goodies on the website.

• Our thanks to Brilliant, for supporting this episode.

We're going to do a silly one today, we're going to talk about a special number.

A2 初級