JAMES CLEWETT: I want to talk about number 41.
41 is a number which crops up in "Garden of Rama." Which,
when I was a teenage boy, was just about my favorite book.
I loved Arthur C Clarke's stuff.
He really enjoyed educating in a really subtle way.
And so in this book--
page 53, I checked--
rather randomly, we suddenly find that we're doing some
sort of quadratic mathematics.
So as a teenage boy, I didn't understand this at all.
I looked at it.
I thought, yeah, that's nice.
And part of me wished I could do that.
But then I moved on.
So what I want to do today is actually take you through this
and explain to you what Arthur C Clarke was talking about,
and why it is kind of cute.
It's the heroine's 41st birthday.
And her husband is a mathematics geek.
And he takes it upon himself to explain to her why 41 is
such a magical number.
Real people don't talk like this.
I work in a maths and physics department.
And even the geekiest mathematicians that I know
don't actually talk like this.
So I just want to reassure you that this is a fictional work.
- Give us an excerpt.
JAMES CLEWETT: OK.
Well, "Therefore, the entire series can generate a simple
quadratic expression.
Take f equals n squared minus m plus 41," he continued,
"where n is an integer between 0 and 40.
That function will generate your entire sequence.
Better still," he laughed, "consider f equals n squared
minus 81 m plus 1681, where n is an integer from 1 to 80.
This quadratic formula starts at the tail end of the
original series, with f equals 1,601, and proceeds through
the sequence, decreasing order first, then reversing itself
to f equals 41, and generates the entire array in increasing
order."
No Nobody talks like that.
- Sorry, I fell asleep.
Are you still going?
[LAUGHTER]
Go on.
JAMES CLEWETT: As a teenager, I wanted to be like this.
I thought that that was really cool.
And so I guess this speaks to me and was a part of, again,
why I became a physicist in the first place.
- Well James, I'm going to make your dreams come true.
JAMES CLEWETT: Thank you.
- Because this is your chance to talk like that and explain
that to us.
[INAUDIBLE].
JAMES CLEWETT: OK.
Let's try and make this make some sense.
And we are going to have to do some math.
But bear with me, OK?
Because I'm going to try and make that math make sense.
OK, so let's start by writing down the equation that they're
talking about.
And that is a simple quadratic formula f equals n squared
minus n plus 41.
Now don't be intimidated.
Because all we're going to do is we're going to say for any
given value of n, what is the value of f?
Let's make a table, shall we?
So if I put in, n equals 0, then I have 0 squared
minus 0, plus 41.
Which is simply 41.
OK?
I put in a value of 1 I have 1 squared, which is 1,
minus 1 plus 41.
So that's simply 1.
If I put in a value of 2, I get 2 squared.
That's 2 times 2 is 4, minus 2 plus 41.
That's 43.
And so on.
I put in 3.
And so I get 9 minus 3, that's 6, plus 41 is 47.
And exactly as Richard Wakefield was saying in the
book, what we're generating here is a
sequence of prime numbers.
Every number that we get for plugging in a value of n from
naught to 40 is a prime number.
Now if I plug-in the value of n of 41, what I get is the
value which is 41 times by 41.
Which is also a rather cute little
symmetry to this problem.
- So you've got this big sequence of prime numbers.
JAMES CLEWETT: Uh-huh.
Which goes up to-- for a value of 40, that goes up to 1,601.
So we've skipped a few there.
But they're all prime numbers, all the way to 1,601.
- You're going to show me something cool now, yeah?
JAMES CLEWETT: Yeah, I think so.
What I'm going to show you is a little something that was
first done by Ulam--
Stanislaw Ulam--
in 1963 while he was bored, sitting in a presentation.
And so he began just to scribble away.
I'm going to start with the number 41 in the
middle of the page.
And I'm going to scribble a spiral of numbers.
So the next number is 42.
54, 57--
- [INAUDIBLE]
JAMES CLEWETT: I'm just making a spiral of numbers.
And at the end of that, I'm just going to get to just a
couple more her.
60, 61--
so what I want to show is how these numbers relate to our
original series of numbers generated from the
mathematics.
OK so our first number in the series, for n
equals 1, we get 41.
And I've put that in the middle.
Nice big circle.
There you go.
Then our second number is 43.
And I've put that one up here.
47, it's down here.
The next number, if I put in n equals 4, the next number I
would find would be 53.
That's up here.
And then we continue and we find 61.
61 is down here.
And so on.
And so what we're discovering is that all of the prime
numbers generated by the quadratic series are found on
this diagonal, the lead diagonal, of what we're
calling an Ulam spiral.
- And every number in that series as you go down is going
to sit along this line here.
Is that--
JAMES CLEWETT: Every number in the series is going to be on
that diagonal line in this spiral.
Now, this is neat because we don't need to start with 41.
We could start with any number.
And I invite you to try and do this at home.
Start with any number.
1, for example.
1, 2, 3, 4, 5, 12, 13--
keep going.
Keep going as long as you feel inclined.
And then go back with a red pen and just draw a circle
around the prime numbers.
3, 5, 7, 13--
I missed one at 11.
So you develop diagonal lines through the system.
And I can draw in other diagonal lines
which would be found--
and if you keep doing this.
- But 9 is not a prime.
JAMES CLEWETT: 9 is not a prime.
No.
There will be gaps.
OK?
But the fact is that there's this amazing pattern that just
emerges out of nature.
Where you see these diagonal lines in clusters.
So if you keep doing this for enough numbers, then you end
up with huge arrays of diagonal lines.
Which I'd like to show you.
Not only on a Cartesian coordinate system where we're
building a square spiral.
But we can plot them on, for example,
an Archimedes' spiral.
And then you get a fabulous pattern which will
look a bit like this.
But 41's really sweet because 41 is the biggest number which
we can put into our original formula, which generates a
series of primes.
We could have used 11, or 17, or 5, or 3, or 2.
And that would generate short series of prime numbers.
41 is the biggest one which generates a series of all
prime numbers.
So this is why our hero in the book likes this series.
He likes it because it generates 40 prime numbers.
And then he can make it mirror itself and generate them all
backwards if he wants to as well.
Because he's a clever chap like that.
- But you couldn't do this with 43?
JAMES CLEWETT: No.
Absolutely not.
This is quite a unique set of numbers that will actually
generate these.
So it was Euler that he first did this in 1772.
So Arthur C Clarke is being a little bit disingenuous when
he claims that Richard Wakefield is
doing it over dinner.
But nonetheless, it's still very, very nice thing.
Euler is a famous mathematician, and geometrist
going back two centuries.
Ulam is the guy who was sitting bored in a seminar one
day and decided to write numbers in a spiral.
As you do, obviously.
But Ulam was an interesting chap in his own right.
He was part of the Manhattan Project.
He holds the patent for the atomic bomb--
or the thermonuclear bomb.
Which is an odd one.
Because if somebody steals your patent, then you don't
need a patent lawyer.
You need a weapons system.
That always amused me.
He also, he was responsible for Project Orion, which was a
project to launch spaceships the size of small frigates
into space using a small series of nuclear explosions.
This was all before they understood
about nuclear fall-out.
He wanted to launch spacecrafts with crews of 200
people on board.
And actually, Orion is still seen as probably the only way
that we have to do interstellar
travel on human lifetimes.
So he was quite a clever guy.
And in his spare time, he drew spirals.
BRADY HARAN: If you haven't seen our other video about
these spirals and prime numbers that was uploaded
around the same time as this one--
and you probably already have seen it.
Because you probably came here from that video.
Well if you haven't seen it, there's a link.
You can go and check it out.
And if you'd like to find out more about James Clewett, how
he became a physicist, why, even footage from his Ph.D.
viva, which is something you don't see every day--
there's also links for that.
Go and have a look.