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

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

It took three volumes.

I've still got it here.

Lots and lots of digits.

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

Now here's a question.

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.

That's not surprising.

There are a lot of digits now.

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

So here's another question.

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.

It's not only yes.

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

This has been proven.

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

And we're about to speak to him now.

We're talking about your favorite subject crimes.

Yes.

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.

23 has no sevens and the digits on dhe.

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.

Similarly, you can ask the same thing.

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.

And James, you cracked this nut, right?

Yeah.

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

And again instead of seven.

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 difficult is always showing it.

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?

So this is an open problem.

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.

Then they'd only be made out of zeroes.

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.

Good point you caught me on.

It's a quite a famous problem, actually.

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.

Three can be all sevens.

All seven sounds good to me.

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

Mono Digit.

I think they're called Rap Unit.

Maybe that's it.

Yeah.

So rep unit crimes.

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.

When I heard that you'd proven this this digit thing, you know, about a missing digit, I thought maybe there would be an exception, like, two or four that they all fall into it, whether they're even or odd or yeah, So there's a few certainties.

The last digital farm number's restricted.

Say, you know that the last digit of a part number certainly can't be an even number.

It can't be five.

Andre, This changes slightly.

How many primes you'd expect that?

Be that Have no sevens or no two's.

You'd expect there to be slightly different numbers because of this last digit phenomenon on dhe simile there.

Some other constraints say to do with some of the digits that in track would be an environment.

No.

But regardless of these small complications, it turns out that there's enough numbers are missing a digit that regardless of what did it is your missing, you can still find times in.

James, I know you've been really involved in work on the twin prime conjecture and gaps between crimes, and these are some of the most famous glamorous problems in mathematics.

Is this a glamorous or important one, or was this almost like just a little curious you found along the way?

Or is this got significance?

So I find it a fun problem to state, but has stated it feels a bit more like a recreational problem.

But what really interests me is the underlying techniques that go into the proof.

The proof was actually quite complicated, and it was developing new tools is to try and get a handle on times.

This was a fine application of these ideas, but maybe the long term goal is that the tools that are developed to get a handle on finds that work in this setting can maybe one day be developed enough to get a handle on times in more complicated settings like the twin find conjecture or something.

We've been working in base 10 so far for discussion, because that's the thing that makes familiar to us.

But you could ask the same question in any off base.

The question gets easier as the basis get bigger.

And so it also follows that you can find infant many crimes that are missing a given digit in any base bigger than equal to 10 on.

Actually, if you look a TTE large enough basis, you can stop avoiding more than one number.

So, yeah, if you want to avoid twos and sevens, we guess that should work in base 10.

But we don't know how to prove it, but we can prove it in a big enough base, so maybe based on 100 would work, maybe needs to be a bit bigger.

I'm not totally sure you could ask the same question as well for smaller bases, but Asli said it gets harder for smaller bases, and so based 10 is at the moment, right at the threshold of our techniques, it's quite fortunate that the threshold is the most natural one fuss.

But one famous conjecture is that there's in front.

You mean Mer's end times?

So these are primes.

Which of the form two to the end minus one.

And these are interesting because often these air very easy to test for whether such numbers of time or not.

So lots of these records for the biggest crimes are all of her former tooth.

The N minus one you can think about.

This is a problem in base two and then in base to any murders in time.

It would just be the digit one repeated, so it would be a rep digit prime in base to or equivalently be a prime that's missing the digits zero in base, too.

So these questions about times missing digits actually relate to some of these really long old standing conjectures about primetime special forms.

Like most times.

James, I've seen you give a lecture about this before, and now you've done a video about here with May, and each time you choose the number seven, you talk about how money problems don't have seven.

Why do you choose seven is your arbitrary example?

I always seem to treat seven, and I don't really know why.

Seven.

Any other number would be just as good.

I sent the read online that when people are asked to choose the one that number they disproportionately to seven on when they choose to did digit numbers, they disproportionately choose 37.

I think so.

They're supposed to be some psychological thing about what seems more random and more like a non specifically chosen number.

But I guess I'm always biased to choose a prime number.

Yeah, often like given silly questions.

When people ask me to choose a line of number, I often choose one just to be difficult, because this is the sort of thing that I find funny.

If you want to find out more about Professor made out, including his favorite number, check out the latest edition of the number file podcast, holding two on the screen and down below, and they'll also be links to more videos with him.