JAMES GRIME: One of the reasons we're fascinated by
primes is that they are quite weird in the way they behave.
On one hand, they kind of feel random.
They are turning up all over the place.
Sometimes you have these long gaps between primes.
And then suddenly-- like buses, you get a couple of
primes turn up at once.
On the other hand, there are things that we can predict
about primes and when they're going to turn up, which is
slightly unexpected that you can do that.
They're not completely random.
One of the first things I want to show you, then, is a nice
easy thing.
So everyone can do this at home.
We're going to write the numbers in a square spiral.
Start with 1 in the middle.
Then you write 2.
But you go around it--
4, 5, 6, 7, 8-- do you see the pattern, then?
It's a square spiral.
12, 13, 14, 15--
it's called an Ulam spiral--
Stanislaw Ulam, he was a Polish mathematician.
And he left Poland just before World War II,
and he went to America.
And he worked on the Manhattan Project.
After World War II, he went into academia.
The story of this spiral is, he sat in a very boring
lecture in academia.
It was in 1963.
And so he's obviously a fan of Vi Hart or someone.
He sat there doodling during this boring lecture.
And he's writing out the numbers.
Let's see, 30, 31, 32.
The next thing he did was start to
circle the prime numbers.
So let's do that.
2 is a prime, and then 3, and then 5, and 7, and 11, 13, not
40, 41, 43, is prime, and so on.
And he noticed, and maybe you can see, these stripes, the
prime numbers seem to be lining up on diagonal lines.
And if you do this larger, if you do more and more numbers,
and you write them out in a spiral, that
tends to be the case.
I've got one here.
This is a big Ulam spiral.
I think this is something huge.
I think this is like 200 by 200.
And so there's 40,000 numbers or something here.
Can you see, though, can you see the stripes?
There's definitely some stripes here,
these diagonal lines.
So prime numbers seem to be lying on diagonal lines.
Or to put it another way, some diagonal lines have lots of
primes, and some diagonal lines don't
have lots of primes.
So you can see the stripes start to form.
BRADY HARAN: Are they continuous stripes?
They look a bit broken up to me.
JAMES GRIME: Yeah, they are not continuous stripes.
But they have more than average number of primes.
So these stripes might be a good place to look for more
primes, bigger primes, new primes.
One thing people might say is, oh, we're just seeing patterns
in randomness.
Those aren't really stripes at all.
It's just the human brain.
See, if you compare it with randomness--
this, the same size, these are random numbers.
And you can see, it's pretty much white noise.
I can't really see any pattern in this.
You can see that that is random.
And you can see that that is something more
than just being random.
BRADY HARAN: These ginormous primes that get found, are
they found on diagonals?
Like this largest prime known, was that on diagonal?
JAMES GRIME: The largest prime known was a Mersenne prime,
which is of the type 2 to the power n minus 1.
It's one less than a power of 2, which is a way to look for
large primes.
It's computationally kind of easier to do.
Perhaps it's not the most fruitful way because they are
quite rare, Mersenne primes.
This might be another way to do it because this stripe
here, this diagonal, has an equation.
This equation is for this one here, this half line, which
means it starts at three and goes off to infinity.
The equation for that is 4x squared minus 2x plus 1.
Let me just try it.
Let's do the first one here.
So if x is equal to 1, yeah, that's 3.
If we tried the next one here, x equals 2, it's 13.
And this one here, that's 31.
And well, best do one more, just to show you what comes
next, 56 plus 57--
is that a prime number, Brady?
BRADY HARAN: 57 is not a prime number.
JAMES GRIME: It's not a prime number.
So the next one isn't a prime number, but 57 would be the
next number on that line.
BRADY HARAN: So that's one of the breaks in our dotted line?
JAMES GRIME: Yeah, so all these lines, the, in fact,
horizontal lines, vertical lines, and diagonal lines,
they are all like this.
All the quadratic equations are like that.
So what we're saying is, some quadratic equations have more
primes on them than others.
And that's the conjecture, actually.
That hasn't been proved.
But that is the conjecture.
It seems to be the case.
So there are lines here that have seven times as many
primes as other lines.
And the best we've found is a diagonal line that has 12
times as many primes as the average.
BRADY HARAN: Cool, has that line got a name?
JAMES GRIME: I can write it out for you.
I think I had it somewhere.
BRADY HARAN: Yeah, I'd love to know what that line is.
The golden line.
JAMES GRIME: This golden line that Brady has now decided to
call it, it's a quadratic equation.
It starts off quite simply again.
But the number you add on is not plus one.
It's plus something huge.
This square spiral is called Ulam's spiral.
But there's one that I like even more.
It's called a Sack's spiral.
And it works like this.
You write the square number in a line.
The square numbers are 1, 4, yeah, that is 2 squared, 3
squared is 9, 16, 25, and so on.
So you write the square numbers in a line.
Then I connect them with what is called an Archimedean
spiral like that.
And then I would the other numbers on that spiral and
evenly space it.
So it goes 1, 2, 3, 4, 5, 6, 7, 8, 9.
And if you mark off the primes for that, I've got this
already sorted out for you, this is the picture you get.
And you can see the relations, you can see the pattern, even
more strikingly, I think.
Look at these curves.
These are the primes.
BRADY HARAN: And obviously, you'll never get a prime along
there because those are the squares.
JAMES GRIME: Those are your squares, that big gap there is
the squares.
So it looks like we have formulas, equations--
some formulas, anyway, that have more primes than others.
So if we can understand these formulas that contain these
rich number of primes, then it would help us solve important
conjectures in mathematics such as the Goldbach
conjecture and the twin prime conjecture.
So prime numbers are not as random as you might think of.
There are equations to help us find prime numbers.
And now I want to show you some equations that help you
find prime numbers.
BRADY HARAN: So we'll have more about ways to search for
prime numbers coming really soon from this interview with
James Grime--
unless you're watching this in the future, in which case this
stuff might already be on YouTube.
But you get the idea.
But, I have a bit of a confession to make.
I've actually recorded some stuff about the spirals and
prime numbers before--
not with James Grime, but with James Clewett.
And I kind of half forgot about it and never got around
to editing it.
This was, like, a year and a half ago.
I went back and had a look, and it was actually really
interesting.
So I've turned that into a video as well.
Now you can wait for that turn up in your subscriptions, in
the next few days, or if you can't wait, you can go and
have a look at it now.
I've made the links available.
The video's already up, so go ahead and have a look.
Thanks for watching.
Plenty more videos, both of stuff I've recorded, some of
it quite a while ago, it turns out, and stuff we've
still got to record.
Really exciting stuff coming soon on "Numberphile," so make
sure you've subscribed.