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  • 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.

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357686312646216567629137 - Numberphile (357686312646216567629137 - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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