JAMES GRIME: So I've got another prime number
generating formula for you.
So this one is more modern.
It was 1947, a mathematician called Mills, and he found
this formula.
He said there exist numbers-- which we're going to call
theta, the Greek letter theta--
where you raise it to the power 3 to the power n.
And you actually then have to round it down.
That gives you fractions, so you actually round it down to
the nearest whole number, which you write like this.
That's the symbol for rounding it down.
There exists a number like this that will always give you
primes for every value of n.
So you might have n equals 1, n equals 2, n equals 3, and
every value is a prime.
The smallest value for theta where this will work is called
Mills' Constant.
I'll write it out for you.
It's called theta, and Mills' Constant is
1.306377883863080690, something, something,
something, something.
So you can see it's not a whole number.
So the value you get when you raise it, it's not going to be
a whole number, and you have to round it down.
Let me just try the first few to show you what I mean.
Let's do n equals 1.
Let's use my constant.
You put it in.
So then it's 3 to the power 1, so it's just cubed.
So what I'm going to do is, for each time, I'm going to
take this constant and then raise it to the power 3
to the power n.
And I'll do that n equals 1, n equals 2, and so on.
So, n equals 1.
3 to the power n is just 3.
So that's going to be theta cubed.
That's the first prime.
And what is the answer?
What is theta cubed?
Theta cubed is 2.229,
something, something, something.
You round it down, so the prime is 2.
Hey!
We've got that is prime.
In fact, every number should be prime.
Let's try the next one.
Let's do n equals 2.
So this is going to be 3 squared.
That's theta to the power 9.
So take the constant to the power 9, and that is going to
be 11.082, something, something.
Round it down, you get 11.
So you get gaps.
It's not consecutive primes, but every one is a prime.
You get 11, that's the next one.
Let's do n equals 3.
So I now need theta 3 to the power 3, 27.
Theta to the power 27 is 1,361.000, something,
something, something.
1,361 is a prime as well.
Let's do n equals 4.
Theta to the power 81.
You get 252,100,887--
BRADY HARAN: Oh, yeah.
That's clearly a prime.
JAMES GRIME: Which is clearly a prime, which is a a prime--
Point something, something, something, but
you round it down.
But you're actually guaranteed to get a prime every time.
BRADY HARAN: Do you know what my conclusion is?
That number's awesome!
JAMES GRIME: I know.
I completely agree.
That number is completely awesome.
So the rock stars of math, so I have pi, and e, and these
golden ratios, and things like that.
It's not nearly as famous.
It's a constant that gives you primes.
Brilliant!
I love it.
BRADY HARAN: Is it proven?
I mean, there's a lot of n's you could
put into that equation.
Is this a proven thing?
JAMES GRIME: Yep.
This is proven.
So you might think then, what's the big deal?
We've got a formula to find guaranteed primes.
How amazing is that?
And the problem is, did you notice how big
the powers were getting?
The powers very quickly become so huge that even computers
can't deal with the problem.
So the next one, n equals 5, is theta to the power 243.
And this is about 1.602 times 10 to the 28.
Once you start getting bigger than that, you can't do it.
The computers can't cope with that.
The other problem is you need to know this constant to a
great accuracy.
Because, what I've written out for you--
let's say if I terminated it there--
isn't accurate enough to tell me what the next prime is.
The next prime actually is something huge.
We'll put it on the screen, I think, instead
of writing it out.
But this was not accurate enough to work it out.
That's a bit of a problem.
You need to know theta very, very accurately.
BRADY HARAN: Is theta rational?
Does theta have an end?
JAMES GRIME: We don't know.
We don't know.
How amazing is that?
We don't know if theta is irrational or not.
That's a cool question as well.
We don't know.
One of the other problems with theta is, at the moment, we
don't know any way to really work it out, apart from taking
one of the Mill primes--
You take one of the Mill primes, cube root it, and
theta is approximately that.
So I'm afraid it's a bit of a circular argument.
We haven't got a good way of working out theta.
You have to know the primes in the first
place to work out theta.
BRADY HARAN: Oh, well I'm not so impressed by it anymore.
JAMES GRIME: I know.
So you can see why it's not so practical.
One of the things that we might be able to do is, if
Riemann's hypothesis is true--
Riemann's hypothesis is a very important hypothesis in
mathematics that hasn't been proven yet, that is a
Millennium prize, and it's related to crimes and how
they're distributed.
If that is true-- (WHISPERING) it probably is true--
If it's true, then we have another way
to find Mill primes.
Using that method, we can now calculate what theta is
probably going to be.
It's been calculated up to about 7,000 digits, but it is
relying on if the Riemann hypothesis is true, which
hasn't been proven. (WHISPERING)
It's probably true.
We have worked out larger Mill primes, and we have worked out
theta to a large number of places.
BRADY HARAN: Our thanks to audible.com for supporting
this video.
They have a huge range of audio books you can listen to.
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to in the car.
And you can download a free book at
audible.com/numberphile.
This is the part where we get to recommend a book, which I
always enjoy.
And today I'd like to recommend "Foundation" by
Isaac Asimov.
The first in a brilliant series of books, probably my
favorite series of books ever.
So go to audible.com/numberphile, get a
free book, and why not check out anything by Isaac Asimov,
but especially "Foundation," or any of
its sequels and prequels.