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.
