Name: 7のない素数 - Numberphile (Primes without a 7 - Numberphile)
Uploaded: 2021-01-14T10:42:15.000Z
Duration: 13 min 15 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

法助動詞 B1

A while ago we made a video about a new world record prime number, millions of digits long.

程度の副詞

Matt Parker even printed out for us, and it didn't take just one volume.

The typeface is tiny, and it's page after page after page of digits.

How many times does the number seven appear in this prime number?

It's like, Let's go to Page one, Line one.

The fifth digit is 1/7 and there are many, many more on the first line and many, many more on every line after that.

In fact, there are well over two million sevens in this prime number.

When numbers get this big, primes become what they become pretty sparse.

Is there a prime number bigger than this one?

So taking more pages, more digits that has no sevens in it, It's almost unthinkable, isn't it, that you could have a number bigger than this?

That's also prime that has no sevens in it, but you know what the answer is.

The answer is there are an infinite number of prime numbers bigger than this one that contained no sevens.

It was proven by a mathematician called James Maine Art of Oxford University.

We're talking about your favorite subject crimes.

Okay, so today we're thinking about the digits of prime numbers on Implicate Flor.

Are there infinitely many prime numbers with No seven's anywhere in their digits?

Or is it the case that there's only finally many crimes that have no second anywhere in the digits?

Say, this might sound like a pretty easy question.

Lots of part numbers that we used Thio have no servants in their digits, so 13 has no seven digits.

When we're looking at small numbers, lots and lots of small numbers have no sevens in their digits.

But when you're talking about infinitely many numbers, you have to talk about really big numbers on dhe.

The vast, vast majority of really big numbers have lots and lots of seventies in their digits and So you're really looking at the only a very small set of numbers that have no sevens.

So one way to make sure how biggest that is, is Thio.

Think about something one over the numbers say, for example, you could choose the prime numbers.

It's a famous result that if you somewhat 1/3, plus one for seven, plus 1/11 and so on that this diverges and say becomes bigger and bigger.

And so in one way, this is saying that there's quite a lot of prime numbers, even though they're quite a sparse set in the whole numbers.

But if I sum up one of the end for N having No.

17 special expansion, then this total sum is less than 100 by this metric is actually quite a small set of numbers and smaller than the set of the primes.

So you might think of crimes is quite rare and difficult to find woes.

Numbers with no seventh anywhere in the decimal expansion has been quite easy to find.

We see them all the time, but actually, when you get to really big numbers, the primes are much, much more common than the numbers with no sevens in their decimal expansion on Dhe.

It's always a big problem for people like me who liked to study the times as thio kind of workout.

If there's infant many crimes in these different odd sets, which are small subsets of the numbers, we all know there are an infinite number of prime numbers.

But you're saying, Do we get to a point where the prime numbers become so big?

And with so many digits, it's impossible to ever find one without seven again.

Yeah, so is it the case that every large enough I'm number always has a least 17?

Somewhere in this, that's more expansion.

You don't expect that most part numbers, they're really big would have lots of sevens in the decimal expansion.

Maybe it could be the case that you can never get ones that are only made up of the digits.

One T 3456890 Once they're bigger than a gazillion or something like that.

Are there in Philly many times that have no zeros in their decimal expansion?

Our infancy, Many crimes have no one's in there desperate expansion.

All the problems are pretty similar to one on dhe simile.

Regardless of whether it's no sevens or no zeroes or no one's, we know that some of us it pickles off those numbers is always bounded by 100 again.

And so again, it's always a fairly small set.

And they expect all really large numbers or, most really large numbers to have lots of sevens and lots of ones and zeros in the desperate expansion.

So I showed that there are indeed infinite many primes that have no seven's anywhere in their desperate expansion.

You can substitute to or three or whatever number you like, regardless, how far you go along the number line so you can go some huge wear long and you choose your favorite point and then sent me somewhere beyond there.

There's gonna be some time, which has no sevens and then another line, which has no threes.

So because there are an infinite number of primes, this fear was intuitively correct because there's always another number.

Did it feel intuitively correct to you before he proved it?

Even though In this sense, the number of numbers with no sevens is quite small.

It's not so small that you would think it's that bizarre that you'd be able to find crimes.

The primes are really difficult to get your hands on, so the question is always can I understand the Times well enough to be able to find them in these somewhat strange, small sets of vintages?

So if there are an infinite number of numbers with no seventh proven well done well done, either an infinite number of primes with no sevens or threes can I can I have two numbers that are missing from my prime?

We don't know this it'll, however, we would certainly expect that it should be infinitely many times with no seconds or threes.

You have to be a bit careful about how far you go along because you can't have times that have no ones twos, threes, fours, 567 or nine schools.

Or you can't have finds that just made out of twos fix on pool.

So there's infinitely many numbers that have just made out of twos.

But all of them are Even so, you're certainly not gonna get any crimes other than to other than two.

Are there infinitely many primes had adjust the number one repeated.

So you certainly can't have it when you have a number of digits, which is a multiple of three.

Because you know that if they're some of the digits adds up to a multiple of three than the number itself is not three, and so it can't be a pint.

Nonetheless, you think that you would have guessed that there should be a infinite number of times that are just only made up of the digit one.

But this would be forced for village it too.

But the only candidates would be one seven, maybe like like a copy.

And even because it can't be all twos, fours, sixes or eight can't be zeros, Captain threes can't be nines, because that would have done exactly then be more.

Yeah, so ones and sevens of the only candidates for What's that called?

So I think once and sevens are the only possibilities for that are you totally have to get on to that one.

Okay, So that one feels a lot harder that somehow just missing one digit in base 10 is right at the limit of the current techniques and even just missing two digits even though that should clearly be possible already feels quite a long way beyond all the techniques that we know moment.

Because thes sets gets smaller and smaller and smaller, you look a bigger and bigger numbers.

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7のない素数 - Numberphile (Primes without a 7 - Numberphile)

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slightly

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