Hi there everyone, you might remember a while ago
we made a video about a new world record prime number, millions of digits long. Matt Parker
even printed it 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 7 appear in this prime number? Let's go, let's go to page 1, line 1.
The 5th digit is a 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
2 million 7s in this prime number. It's not surprising.
There are a lot of digits. Now when numbers get this big primes become - well 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 do 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 contain no sevens. This has been proven,
it was proven by a mathematician called James Maynard at Oxford University
and we're about to speak to him now.
(Brady: We're talking about your favourite subject) - Primes! - (Brady: Yes)
Okay. So today we're thinking about the digits of prime numbers and, in particular, are there infinitely many prime numbers with no
sevens anywhere in their digits? Or is it the case that there's only finitely many primes that have no sevens anywhere in their digits?
So this might sound like a pretty easy question, lots of prime numbers
that we're used to have no sevens in their digits. So 13 has no sevens in the digits,
23 has no sevens in the digits, and
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 and
the vast, vast majority of really big numbers have lots and lots of sevens in their digits,
and so you're really looking at only a very small set of numbers that have no sevens.
So one way to measure how big a set is is to think about summing 1 over the numbers.
So for example, you could choose the prime numbers,
it's a famous result that if you sum up 1 over 3, plus 1 over 7, plus 1 over
11, and so on; this diverges and so, becomes bigger and bigger and bigger.
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 their whole numbers.
But if I sum up
1 over n,
for n having no sevens in its decimal expansion
then this total sum is less than
100. By this metric it's actually quite a small set of numbers and smaller than the set of the primes.
So you might think of primes as quite rare and difficult to find, whereas numbers with no sevens anywhere in their 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.
And it's always a big problem for people like me who like to study the primes as
to - kind of work out if there's infinitely many prime' in these
different odd sets which are small subsets of the numbers.
(Brady: We all know there are an infinite number of prime numbers) - Yep
(Brady: 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 a seven again?)
Yeah, so is it the case that every large enough prime number always has at least one seven somewhere in its decimal expansion?
You'd certainly expect that most prime numbers that are really big would have lots of sevens in their decimal expansion,
maybe it could be the case that you can never get ones that are only made up of the digits
one, two, three, four, five, six, eight, nine, zero;
one step bigger than a gazillion or something like that. Similarly, you can ask the same thing,
are there
infinitely many primes that have no zeros in their decimal expansion?
Are there infinitely many primes that have no ones in their decimal expansion?
All the problems are pretty similar to one another and similarly
regardless of whether it's no sevens or no zeros or no ones,
we know that the sum of the reciprocals of those numbers is always bounded by 100 again.
And so, again,
it's always a fairly small set and so you expect all really large numbers, or
most really large numbers, to have lots of 7s and lots of ones and lots of zeros in their decimal expansion.
(Brady: And James, you cracked this nut?)
Right. Yeah
So I show you that
there are indeed infinitely many primes that have no sevens anywhere in their decimal expansion and again, instead of seven
you can substitute two,
or three or whatever number you like. Regardless how far you go along the number line,
so you can go some huge way along and you choose your favourite point, and then certainly somewhere beyond there
there's going to be some prime which has no
sevens and then another prime which has no threes, and so on. - (Brady: Because they're an infinite number of primes,)
(this feels intuitively correct doesn't it? You know, because there's always another number.)
(Did it feel intuitively correct to you before you 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 primes.
The difficult is always showing it. The primes are really difficult to get your hands on and so the question is always,
can I understand the primes well enough to be able to find them in these somewhat strange
small sets of integers? - (Brady: So if there are an infinite number of numbers)
(with no sevens. Proven, well done.) - Yeah - (Brady: Well done)
(Are there 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 at all,
however
we would certainly expect that there should be infinitely many times with no sevens or threes.
You have to be a bit careful about how far you go along, because you can't have primes that have no
ones, twos threes, fours, fives, six, sevens or nines 'cause then they'd only be made out of zeros. Or, you can't have primes
that're just made out of twos for example, so there's infinitely many numbers that are just made out of twos,
but all of them are even so you're certainly not going to get any primes in there - (Brady: Other than 2) - Other than 2!
Good point you caught me. And it's a quite a famous problem
actually, are there infinitely many primes that are just 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 the sum of the digits adds up to a multiple of three then and the number itself is a multiple
of three, and so it can't be a prime. Nonetheless you think that - you would have guessed that
there should be a infinite number of primes that are just only made up of the digit one,
but this would be false for the digit two - (Brady: But the only candidates would be one...)
(seven maybe? Like, like it can't be an even because it can't be all twos, fours, sixes or eights.)
(Can't be zeroes, can't be threes, can't be nines because that would add up to - ) - Exactly, then it'd be a multiple of three
(Brady: Could it be all sevens?)
All sevens sounds good to me. Yeah. - (Brady: So ones and sevens are the only candidates for,)
(mmm, what's that called mono digit?) - I think they're called repunit? Maybe? - (Brady: That's it, yeah)
Yeah, so repunit primes. So I think ones and sevens are the only possibilities for that.
(Brady: Oh, you totally have to get onto 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 at the moment;
because these sets get smaller and smaller and smaller too you look at bigger and bigger numbers.
(Brady: 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 but they all fall into it, whether they're even or odd or?)
Yeah, so there's a few subtleties. The last digit of a prime number is restricted
so, you know that the last digit of a prime number certainly can't be an even number. It can't be a five,
and so this changes slightly how many primes you'd expect there to be that have
no sevens or no twos. You'd expect there to be slightly different numbers because of this last digit phenomenon. And similarly
there are some other constraints, say to do with the sum of the digits, that interact with being a prime or not.
But regardless of these small complications it turns out that there's
enough numbers are missing a digit that, regardless of what digit it is that you're missing, you can still find primes in them.
(Brady: James I know you've been really involved in work on)
(the twin prime conjecture, and gaps between primes, 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 curio you found along the way? Or has this got significance?
So I find it a fun problem to state but as 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 to try and get a handle on primes. This was a fun application of these ideas
but maybe the long-term goal is that the tools that are developed to get a handle on primes that work in this setting can
maybe one day be developed enough to get a handle on primes in more complicated settings like the twin prime conjecture or something.
We've been working in base 10 so far for our discussion, because that's the thing that's most familiar to us,
but you could ask the same question in any other base. The question gets easier as the bases get bigger.
And so it also follows that you can find infinitely many primes that are missing
a given digit in any base bigger than equal to ten.
And actually if you look at large enough bases, you can start avoiding more than one number.
So yeah, if you want to avoid twos and sevens,
we guess that should work in base ten, but we don't know how to prove it,
but we can prove it in a big enough base. So maybe base 100 would work. Maybe it needs to be a bit bigger
I'm not totally sure. You can ask the same question as well for smaller bases
but as I said, it gets harder for smaller bases and so base ten is, at the moment, right at the threshold of our techniques.
It's quite fortunate that the threshold is the most natural one for us,
but one famous conjecture is that there's infinitely many Mersenne primes.
So these are primes which of the form
2 to the N, minus 1 and
these are interesting because often these are very easy to test for whether such numbers are prime or not,
so lots of these
records for the biggest primes are all of the form 2 to the n minus 1. You can think about this as a problem
in base 2, and then in base 2 any Mersenne prime would just be the digit 1 repeated.
So it would be a repdigit prime in base 2, or equivalently
it'd be a prime that's missing the digit 0 in base 2. So, these questions about primes missing digits
actually relate to some of these really long old
standing conjectures about
primes of special forms like Mersenne primes.
(Brady: James I've seen you give a lecture about this before, and now you've done a video about it here with me, and each time)
(you choose the number 7. You talk about how many primes don't have a 7.)
(Why do you choose 7 as your arbitrary example?)
I always seem to choose 7 and I don't really know why I choose 7,
any other number would have been just as good. I've certainly read online that
when people are asked to choose a one digit number, they disproportionately choose 7, and when they choose two digit numbers
they disproportionately choose 37 I think.
So there's 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, I often like - given
silly questions when people ask me to choose a random 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 Maynard, including his favourite number check out the latest edition of the Numberphile podcast,
I'll link to it on the screen and down below. And there'll also be links to more videos with him
so check out those links as well.