THE NUM8ER MY5TERIES –
Presented by Professor Marcus du Sautoy
LECTURE 1: THE CURIOUS INCIDENT OF THE
NEVER-ENDING NUMBERS
Broadcast on: 25th December 2006
Part 1
Marcus
Hello and welcome to the 178th Royal Institution Christmas Lectures. This year’s
lectures are called The Number Mysteries. Over the next few days I’m going to
show you how mathematicians have helped us understand some of the big
mysteries of the universe. But I’m also going to explain to you how mathematics
can make you rich.
Mathematics has solved a lot of the mysteries of the universe, but there’s still a lot
of problems that even the cleverest of mathematicians cannot understand. And a
businessman in America has offered one million dollars for anybody who can solve
any of these mysteries. Now why is this businessman offering money for
mathematics problems? Well, he realises that mathematics can help us answer
problems of science, technology, the economy. It can even help us to save the
planet. So each one of the next five lectures is going to be an introduction to how
to win this million dollars. And they don’t come any bigger than our first problem:
The curious incident of the never-ending numbers.
Now you might think we know everything there is to know about numbers, after all
we’ve been studying them for several thousand years. Surprisingly, there are still
a lot of mysteries about these numbers as they whizz off to infinity. The most
enigmatic numbers of the whole of mathematics are the prime numbers. Now, a
prime number is a number which is only divisible by itself and one – like 7, or 11 –
and they’re my favourite numbers in the whole of mathematics. My life is full of
prime numbers: I live in a prime number house, my car registration number is
prime, I even persuaded my football team, Recreativo Hackney, to change its
football kit and now we all play in prime number shirts. It transformed our season,
in our prime number shirts we managed to get promoted to the first division.
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So how did mathematics help us to find the prime number shirts that our football
team play in? Well, if that million dollar explosion wasn’t enough for you, we’ve
got some more explosions for you in the Christmas Lectures. Here’s an explosive
demonstration to help us to find the primes in our football team from 1 up to 50.
For this demonstration I’m going to need four volunteers. So who’d like to help me
to find the primes in the numbers from 1 to 50? OK, if you could come up for me
sir? If you’d like to stand here please. And you’d like to come up? And we need two
this side. If you’d like to come up? We’re going to have some detonators. And let’s
have you, yes you’re desperate to come up. Come on up then. I’ve got some
detonators, now please don’t let the detonators off too early else we’ll bring the
lecture theatre down. We need safety glasses as well, because we’re going to blow
some things up. So you’re going to stand behind this one. Have you got some
glasses for me Andy? Just in case things really blow up again. Ok, we’re going to
find some way to burst all the balloons which are not prime numbers up here
behind me. So, I’m going to get my four volunteers to help me. We’re going to
start with the first number first: the number 1. Do you think I should burst this
because it’s not a prime? Or should I leave it up there because you think it is a
prime? Now you should find underneath your seats some cards to vote with.
You’ve got a card with a red and a white side on, so I want you to show the red
side if you think I should burst this because it’s not a prime, and the white side if
you think I should leave it up because you think it is a prime. OK if you’re ready to
decide whether I should burst it or not? Let’s take our vote now. Ooh, it’s pretty
equally divided, quite a lot of red over here and at the back. So, mostly, you seem
to think actually it’s probably not prime number and I should burst it, but, actually
I’m not sure this is divisible. What am I going to divide it by? Intriguingly, if I’d
been giving the first Christmas Lecture in 1825, when they were first given. Well
mathematicians used to think this was a prime number, and I should leave it up
and I shouldn’t burst it. After all it isn’t divisible. But, in 2006 we think about the
primes slightly differently, they’re building blocks: you don’t really get anything
new when you multiply by 1. So all of you who showed the red side are right, I
should burst this balloon because in 2006 we say it’s not a prime number. So let
me burst it. (BANG.) Next, here is the number 2, that’s the first prime number.
Anything in the 2 times table isn’t a prime, so I’m going to knock out all the even
numbers, my first volunteer is going to help me do it. So what’s your name sir?
Robert
Robert
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Marcus
Robert, is going to knock out everything in the 2 times table, except for 2. People
are already putting their fingers in their ears, just in case of the explosion. Three,
two, one, go. There we go, fantastic; we’ve knocked out half the balloons there.
Oops, we knocked out 31; we shouldn’t have knocked that one out, that’s actually
a prime, but well done. Ok, our second volunteer, we move along here, there’s the
number 3 is the next prime along so I’m going to knock out everything in the 3
times table which hasn’t already been knocked out. So what’s your name?
Ella
Ella.
Marcus
Ella’s going to knock out everything in the 3 times table. Three, two one, go. We’re
clearing things very nicely here. Four has already been knocked out, so the next
prime number is 5. My volunteer here is going to knock everything in the 5 times
table out. So what’s your name sir?
Harry
Harry.
Marcus
Harry, ok, you’d like to knock out everything in the 5 times table, let’s count him
down: three, two, one. Very good, only two left there actually, so we move along,
we’ve already knocked out 6, that’s not a prime. The next prime we find is 7, so I’m
going to knock out every seventh balloon and my last volunteer here is going to
help me do that. So what’s your name sir?
Tim
Tim.
Marcus
Tim is going to knock out everything in the 7 times table, let’s count him down:
three, two, one, plunge. Right, only one left there, was pretty pathetic, but now
we’ve been left with all the primes in the numbers from 1 to 50. So let’s give our
volunteers a great round of applause for finding the primes. Thank you, go back to
your seats, yeah, you can put your glasses on the side there.
Well, believe it or not it was actually a librarian in ancient Alexandria who came up
with that way for finding the primes. With just four detonations we’ve actually
found all the primes between 1 and 50, that’s enough for my football team, but the
primes they go on forever. As I look more and more through these numbers I’m
going to find more and more primes scattered in amongst them. And popping
balloons becomes a little bit inefficient, so I’d like to find some pattern in these to
help me to find more of the primes. The trouble is if you look at these numbers,
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there doesn’t seem to be much pattern at all. Here’s the number I play in, 17,
which is followed quickly after by 19, and I come down here, 23, then I’ve got a
massive gap of six numbers none of which are prime. I’ve got 41 and 43, which are
rather special primes. They are the closest that two primes can be together; for
the one in the middle is always going to be even. So we actually call these twin
primes. I have twin daughters actually, but my wife wouldn’t let me call them 41
and 43 – that’s my secret middle names for them. But this character about the
primes where you get some very close together and then big gaps, carries on, as
you look through the primes, bigger and bigger, seems to have this very
unpredictable behaviour.
As a mathematician I would love to find a pattern in these numbers. Now when I
say I’m a mathematician most people would think that what I do long division to
lots of decimal places. Actually, what I do all day is trying to look for patterns. A
mathematician is a pattern searcher. I am desperate to try and find some pattern
to help me find these primes. The reason I’m so obsessed with these numbers is
not because they helped my football team to get promoted (although that was
very nice of them), it’s because they are the building blocks of my subject. And I
will need a volunteer to try and help me to explain why the primes are building
blocks, yeah, if you would like to come up over here. So, I’ve got a number for you
here and we’re going to try and see, first of all, whether it’s a prime number. So if
you’d like to stand the other side of this number here. You come here, ok and
what’s your name?
Morgan
Morgan.
Marcus
Morgan. Morgan is going to try and see whether this, first of all, is a prime
number, do you think this is a prime number, 105? Is it divisible by anything? Yeah,
it’s divisible by 5. Anything ending in 5 is probably not going to be prime number,
except for 5. So we can divide this by 5 and we get down to, now, if you want to
take this side, we’re going to spin this side, here we get to 21 times 5. 21, is 21 a
prime number? Or can I divide it?
Morgan
Prime number.
Marcus
You think it’s a prime number? Or do you think it might be divisible?
Morgan
Divisible.
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Marcus
It is divisible, what’s it divisible by? Do you know?
Morgan
Seven.
Marcus
Seven, exactly, it’s 3 times 7 Let’s turn again and we get, you pull the 3 towards
you, we’ll see a little multiplication sign appearing. So in fact we’ve divided this
into 3 times 7 times 5. Now, can we divide any of these other numbers? Or are
they prime numbers?
Morgan
Prime numbers.
Marcus
They’re prime numbers, they’re indivisible numbers. We can see them up on the
board here, so these we can’t divide any further, these are the prime numbers
which built the number 105. So let’s give a great hand, round of applause for
finding the primes to build 105.
Now we can actually do this with however large a number we have. We can keep
on dividing it, dividing it and dividing it until we get down to the primes which built
that number. So for me these primes are a little bit like the atoms of my subject.
They’re a bit like, the chemists have this thing, the periodic table, has all their
atoms from which you can build molecules, so for example the first atom. Does
anyone know the first atom? Yeah, hydrogen, hydrogen is the first atom in the
periodic table, so you can build things with all these atoms. We’ve got some
hydrogen for us, here Andy’s got some hydrogen in this balloon and we’re going to
combine this with a little bit of oxygen, we’ve got some oxygen around here. So we
combine hydrogen and oxygen we get, like these, combining these numbers we
get something new. I’m going to, going to hide behind here actually, yeah, this is
why I became a mathematician because I hate explosions, so, put your safety
glasses on Andy. Wow, so hydrogen and oxygen, what does hydrogen and oxygen
make? Water! Ok, but where is the water?
Andy
There’s only a tiny bit.
Marcus
Ok, it’s all got vaporised. But you combine hydrogen and oxygen and you get
some water, well, thank you Andy for (LAUGHS) another explosion. Well these are
a bit like the hydrogen and oxygen of the world of mathematics, in a sense this is
my periodic table, with all the atoms which I can build numbers from. So from the
primes I build numbers, from numbers I get mathematics, from mathematics I get
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the whole of science. So these primes are sitting down there at the foundation of
my subject, in a sense now we can see why one shouldn’t be a prime because, if I
multiply by any of these, I get something new. If I multiply by 1, I get nothing new.
So the most fundamental numbers in mathematics. Now, I would love to find a
pattern behind these numbers, and, after the break, we’re going to embark on our
search for a pattern to help us find these prime numbers. But before we go, here’s
a question for you to think about during the break - who were the first to discover
the primes? Was it a) the Ancient Greeks, b) insects, or c) the Ancient Chinese?
Come back after the break to find out.
Part 2
Marcus
Welcome back to The Number Mysteries. We saw, in the first part of the lecture,
that the prime numbers are so important, primes like 13 and 17, because they are
the building blocks of the whole of mathematics. I asked you, who were the first to
discover the primes, well, surprisingly enough it wasn’t mathematicians at all, but
a curious little insect which lives in the forest in North America. Now this insect
has a very curious life cycle. It hides underground doing absolutely nothing for 17
years, then, after 17 years, these cicadas emerge, en masse, into the forest. Now
they sing away, you can probably hear a few of them there, you need to multiply
this by about a hundred thousand of these cicadas, actually the sound of these
cicadas is so loud that residents have to move out of the area, it’s unbearable.
These cicadas party away, then after 6 weeks they all die, and the forest goes
quiet again for another 17 years. Now that’s an amazing feat, to be able to stay
underground for 17 years and then to appear, but is it just a coincidence that 17 is
a prime number? Well we think not because there’s another species in another
forest where they hide underground for 13 years, another species where they hide
underground for 7 years. So what is it about the primes, they’re all primes, 7, 13,
17, what it is about the primes that are trying to help these cicadas in some way?
Well we’re not actually too sure, but we’ve got a hypothesis. We think that maybe
there was a predator that also used to appear periodically in the forest and the
predator tried to time its arrival to coincide with the cicada and they would gobble
the cicada up. Now, we’re going to test out this thing about primes and cicadas to
see whether it’s right, and first of all, to do this experiment I need to change the
lecture theatre into a nice North American forest. So let’s change our lecture
theatre into a forest, that’s lovely. On the floor here I’ve got numbers from 1 to 42,
they represent years in the forest and I’m going to try and see what happens if I
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set some predators, I’m going to have some poisonous wasps on every sixth year
in the forest. So, I’ve got some predators already primed to come out so if you’d
like to come and stand behind the, pick up your helmets, these are poisonous
wasp helmets, so if you’d stick those on your head. Now, these predators are
appearing in the forest every 6 years so the first one is year 6, the next one year
12, 18, all the way through to the last one at 42. Now, what I want to find out is
what happens to a cicada in this forest where there’s a predator. So I need a
volunteer to be a cicada who’s going to appear every 9 years in the forest, yeah if
you’d like to come up sir. I’ve got a cicada helmet for you, here you go, right, what
you’re going to do is start down this end of the forest. Ok, what’s your name sir?
Max
Max.
Marcus
Max, if you’d like to put this cicada mask on your head. I think the best thing is to
put it under here; there we go. Now this cicada is going to appear every 9 years in
the forest, so, not a prime number. So let’s see what happens to him every 9
years. Now, it’s likely you’re going to get killed at some point, so I want a really
dramatic death when you get killed because there’s going to be a prize for the
best death, are you ready? Now, you’re staying underground, so we’re going to
kneel down, and you’re going to creep along the forest and you’re going to appear
every 9 years. So, here we go, first 9 years up he pops, parties away because there
are no predators to get him, so back down you go. Right, where’s the next 9, 9
plus 9 comes to 18, but unfortunately the predator’s here in year 18, so what do
you think, hmm, pretty good, yeah. Let’s sweep him up, this dead cicada. If you’d
like to move into the middle there, and you’ve got to stay dead, go on you can die
again, alright, there you go, excellent. Ok, so this cicada appearing every 9 years,
not a prime, didn’t do too well in the forest, got killed at year 18 when he met a
predator. So now I want a volunteer to be a prime number cicada, so who’d like to
be a prime number cicada, yeah would you like to come up and be a prime number
cicada, up you come. Here we go and we’ve got a cicada helmet for you, so if
you’d like to put that one on. And what’s your name?
Catherine
Catherine.
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Marcus
Catherine is going to appear every 7 years in the forest, so I want you to kneel
down, if you crawl this side of the numbers so we can all see them. She creeps
along, appears in year 7, up you come, pretty good, no predators, down you go, so
she’s going to appear again. Now you might think that appearing every 7 years, so
she got to 14, the cicada knows her 7 times table, back down again you go. You
might think she’s appearing more often so she might actually get caught more
often, but although she’s appearing more often she seems to be avoiding all of the
predators. By keeping out of sync, using a prime number, here she is, yeah, down
you go 28 that was, excellent, next time is 35, 35 is the next one in the 7 times
table, avoiding all of these predators. Come on, you’re trying to get that cicada,
but you can’t get her until finally she appears at year 42, which is the first time we
have a year which is divisible by both 6 and by 7. Oh that was a pretty good death
I think. Ok so the predators couldn’t get this one, because by using a prime
number she could keep out of sync and avoid all the predators. So I think we’ll
give both our cicadas a round of applause, I think they both deserve a present.
Here we go, so you can take that back, I think it’s some jellied worms or
something like that. I don’t know whether cicadas eat worms, but there you are
and you can go back to your seats, great. I’m afraid the predators went back pretty
hungry after that because they didn’t get much to eat, but you see the cicada that
appeared every 7 years seemed to survive much better than the one that
appeared every 9 years. So the prime numbers seem to be the key to the
evolutionary survival of these cicadas, and it seems like a competition developed
in some forests where they got to 7 in our forest, but then in another forest it went
all they way up to 13. In one forest they got up to 17 years, and so I think the
predator couldn’t find the primes. He was a stupid predator, didn’t know his prime
numbers and died off, and we’ve been left with this cicada with a prime number
life cycle.
Now for cicadas we’re using evolutionary tactics to find their primes, but that isn’t
a very efficient way to do things and, you know we used the balloons at the
beginning, that’s also not very good at finding primes. What I’d really love is some
sort of formula to try and help me find the primes. Mathematics is full of formulas
to help us to predict what is going to happen next, and we’re going to do a little
warm up to try and find a formula to try and predict a number sequence.
I’ve got some volunteers at the back; don’t put your balls up already. They’ve got
some numbers on a ball and the challenge for all of you is to try and see whether
you can spot what’s the next number in the sequence. Can you spot the pattern in
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the following example? So are you ready for your number challenge, let’s show
your numbers. We’re starting with 1, then 3, then 6, then 10, then 15, now this boy
here has got the number wrapped up in a parcel so we’ve got to find out what is
the number in that sequence. A lot of you seem to think you know what it is, but
here’s a little hint to see, so what do you think it is sir?
Speaker
21.
Marcus
Shall we open the parcel to see whether it really is 21 – if you can hold your
numbers up, what’s the next number in the sequence? It is in fact number 21, very
good. Now if you haven’t got why 21 is the next number in the sequence, here’s a
little hint for you, it’s a little Christmas song. Today’s Christmas day, the first day
of Christmas, and my true love, who’s sitting somewhere in the audience tonight,
actually gave me a present, she gave me a partridge in a pear tree. Now tomorrow
for Boxing Day I’m expecting a few more presents, I’m expecting to get two turtle
doves. In fact we’ve got sort of ninja turtle doves here and a partridge in a pear
tree. On the third day of Christmas I’m hoping she’s going to send me three French
hens, running along the bottom, two ninja turtle doves and a partridge in a pear
tree. The first day I got 1 present, the second day 3 presents, on the third day I got
6 presents, so if you’d like to hold your numbers up again we’ve got 1, 3, 6. These
are in fact the numbers, the number of presents I’ll get in the 12 days of Christmas.
And it helps us to find out how to get the next number, because on the fifth day I
got 15 presents, on the sixth I’m going to get those 15 presents again, plus another
6 more, so you have to add 6 to get the next number in the sequence. So here the
number of presents on the first few days of Christmas. But what if I wanted to try
and work out the number of presents I’d get on the hundredth day of Christmas.
So over here we’ve got a parcel for the number of presents, don’t open your parcel
yet, we’re going to try and work out how many presents I’m going to get on the
hundredth day of Christmas. There are several ways I could do this. I could add up
all the presents, 1 plus 2 plus 3 all the way up to 100. Now that’s a really stupid
way and Andy over here has pulled the short straw. He’s going to add up all the
numbers from 1 to 100, but I’m going to show you why a formula can help us to get
it much more quickly. So off you go Andy, you’re going to need a bit of time to do
this. Ok, who’d like to try and help me find a formula to calculate the number
we’ve got over here? Yeah, if you’d like to come up here? Right, we’re going to
calculate it over here, we’re going to find a little formula. So if you want to come
over to my presents, we’re going to use these presents to try and find this formula,
ok. So what’s your name?
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Jemma
Jemma.
Marcus
Jemma, we’re going to try and calculate the number that’s on that board, and
we’ve got to beat Andy who’s calculating away over there. Are you ready? What
we’re going to do is start with working out the number on the third day of
Christmas. Now, actually mathematicians call these the triangular numbers, not
the 12 days of Christmas, because it’s a nice way I can arrange the presents in a
triangle, I’ve got one here, two here, three more here. And we’re going to use this
triangle to create a formula to calculate that number really quickly. I want you to
help me take off this triangle we’ve got down here, so If you pull, there we go. So
you’re going to hold this triangle. Now I’ve got two copies of this triangle and I’m
trying to work out how many boxes are there in this triangle, but, if you see, we
can lock this together and actually make a rectangle. Now calculating things in
rectangles is very easy, so I’ve got 3 down this side and I’ve got 3 French hens plus
one, 4 down this side, so, 3 times 4 gives me?
Jemma
Twelve.
Marcus
Twelve, exactly, that’s the number of presents in this whole rectangle, but that’s
twice the number of presents I’m trying to calculate so I need to dived 12 by 2 to
get?
Jemma
Six.
Marcus
Six, and sure enough there’s 6 presents. Ok, let’s see how Andy’s doing, where
have you got up to Andy?
Andy
I’m about to do 26.
Marcus
Twenty-sixth day of Christmas, all right, so we’ve still got a bit of time. Now, we’re
going to use the same principle to calculate the hundredth day of Christmas. Let’s
stack up our presents, pretend we’ve got 100 presents stacked up on this side,
now we’re going to take a second copy of the rectangle. I’ve got 100 days of
Christmas down this side, now I’ve got 100 plus 1 along the bottom, so how many
squares are there? It’s 100 times 101, so can you do that one? Ten thousand, one
hundred.
Jemma
Yeah.
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Marcus
Ok, so that’s 10,100 boxes here, but again we’ve got twice as many as we want. So
we need to divide by 2, ok, 10,100 divided by 2, see I’m afraid she’s got the really
difficult calculation.
Jemma
Five thousand and fifty.
Marcus
Five thousand and fifty, let me go and have a look, hold on to that triangle here.
Let’s see whether 5,050 is the right answer, so if you could unwrap the parcel, this
is the number of presents I’m going to get. Am I going to get as many as 5,050
presents on the hundredth day of Christmas? If you could hold up your ball, yes,
5,050, well done we managed to calculate the formula, and where has Andy got
to?
Andy
Thirty nine.
Marcus
Thirty-ninth day of Christmas. Thank you very much you’ve managed to calculate
very fast, using this formula for the hundredth day of Christmas. So let’s give her a
big round of applause for calculating the number. So shall we let Andy off finishing
the calculation? Yeah I think so, ok.
The wonderful thing about this is that actually I’ve found this formula that I can
use to calculate say the millionth day of Christmas, how many presents I get on
the millionth day. I just take a million times a million plus 1, and divide it by 2.
Using this formula I can find as big a triangular number as I want to, so the
triangular numbers we can understand, but it’s these prime numbers that we want
to really understand. I’d like to find a formula for really big primes.
Now it’s not just mathematicians and my football team, Recreativo Hackney, who
are interested in finding big primes, actually prime numbers are now used as the
building blocks for internet security. Every time you send your credit card securely
across the internet you’re actually using very big primes to keep it secret, so it’s
also the internet that want to find these big primes and they’d love a formula too.
In the fourth lecture in this series I’m going to explain a little bit more about codes
and how primes are used to build codes, but for the moment let’s try and find
some way to find a formula to find big primes. After the break and I’m going to
explain to you a formula developed by a French monk called Marin Mersenne 350
years ago. In order to understand the formula that Marin Mersenne developed I’d
like you to welcome Mr Low from deepest Dagenham. Give a big Royal Institution
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welcome to Mr Low. Mr Low is going to make us some dragon noodles. He’s got us
dough here, if you’d like to start preparing the dough. He needs to sort of make
the dough nice and relaxed and soft, and what he’s going to do is he’s going to
show the power of doubling in mathematics to make really big numbers. During
the break he going to try and double this piece of dough 7 times, so it’s going to
get longer and thinner, longer and thinner. And, during the break I want you to try
and estimate how long you think this noodle is going to be that Mr Low’s going to
make for us. Do you think that the noodle will get from here to the back of the
lecture theatre a), or b) do you think it will get to the River Thames, 200 metres
away, or c) do you think it will stretch from here to the Millennium Dome 4 miles
away? We’ll come back after the break to find exactly how big Mr Low’s noodle is
going to get.
Part 3
Marcus
So far we’ve seen the prime numbers are really unpredictable, which makes them
really difficult to find. But my football team, Recreativo Hackney, who all play in
prime number shirts, have got big plans for our next season. We want to make
some new signings, and we’re going to need to find some really big prime
numbers for them to play in. Now in this part I want to explain to you a formula
which a monk, 350 years ago, found for finding record breaking primes, and it
relates to what Mr Low is doing here now. Mr Low is getting ready to do his last
doubling, is that right?
Mr Low
Yep.
Marcus
Here we go, great, they’ve become really long and thin, and here we have dragon
noodles by Mr Low, fantastic. Now in 2 minutes Mr Low has managed to double
that 7 times, but already that’s produced enough noodles to stretch from here to
the river Thames 200 metres away. In fact by doubling things you can get really
large numbers, there should be about 128 noodles, I think in here in total.
Mr Low
Yes.
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Marcus
Ok, let’s give Mr Low a great applause for making an exceedingly long noodle, so
thank you Mr Low. Well, in the challenge for finding really long and big numbers,
that’s not terribly big. If you want to find out the power of doubling, it’s not
noodles but in fact rice you need to turn to. Now I’m going to need a chess board
to explain how doubling with rice helped Marin Mersenne and modern
mathematicians to find some of the biggest prime numbers we’ve so far
discovered. I’m going to need a volunteer to help me find this, ok why don’t you
come up sir. Now we’ve got a little costume for you, so if you’d like to come over
here. Here’s our costume, and here’s some rice, ok, so what’s your name sir?
Ben
Ben.
Marcus
Ben is going to put on a crown and I’ll explain the significance of the crown in a
little bit. And now Ben what I want you to do is to put 1 grain of rice on the first
square of the chess board down here, and then you’re going to put 2 on the next
one, 4 on the next, and doubling up each time. I’m going to set him off so you can
kneel down and off you go. Right, there’s a little story behind this demonstration,
legend has it that chess was invented by a mathematician in India, and the King of
India was so pleased with the new game that he promised the mathematician
anything he wanted. So the mathematician thought for a while and he said ‘I’d like
1 grain of rice to be put on the first square of the chess board, 2 on the second, 4
on the third and doubling up each time’. Now, the king of India thought he’d got
away extremely lightly, but he was in for a surprise, he didn’t realise the power of
doubling. So let’s see how far you’ve got Ben, what are you up to?
Ben
I’m on the fourth square.
Marcus
Fourth square, ok yeah, it looks a pathetic amount of rice, the king must have
thought, you know, what a small prize for such a wonderful game. But look what
happens as we get further through the chess board, we’re going to stop you there
and get you to put the number of grains of rice there would be on the sixteenth
square of the chess board. That’s going to be about a kilo of rice, so if you’d like to
pile a kilo of rice down here onto the sixteenth square, great, let’s pile it up. Still
not too bad, the King of India could go down to the supermarket and buy that; pile
it on, great, well done. Right, now we’re going to move to the twentieth square,
how much have we got by the twentieth square? Here you go, alright, leave it
there for him, Ben’s got to do this, he’s the king of India. We’re going to have
about 2 sack loads, so this is the twentieth one, put it all down for me, excellent;
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keep pushing. So this mathematician is beginning to get more rice than the king
thought he was going to have. Ok, Ben, we’ve got to come to the sixty-fourth one,
the last one. So how many grains of rice do you think are going to be on the chess
board by this time? It’s a huge…
Ben
Quite a lot.
Marcus
Quite a lot. Here’s exactly how much, this is the number of grains of rice, in total,
on the chess board: it’s 18 billion, billion grains of rice. Now that’s a lot of rice, in
fact if we piled that amount of rice up it would stretch from one side of the M25 to
the other, and it would cover all the buildings in London. In fact this is more grains
of rice than can be grown on the surface of the planet, I think, in about a thousand
years. So, unsurprisingly the king wasn’t able to give the mathematician what he
wanted and he had to part with his fortune instead, one way that maths can make
you rich. Let’s give Ben a great round of applause, now keep this number here.
Great, thank you Ben.
Now, this is a very large number, but it’s not a prime number, so how can we use
the rice on the chess board to make large, and also prime, numbers? So take your
number away, divisible by 5, don’t want that one. Marin Mersenne realised that
actually when you looked at the prime number squares on the chess board
sometimes you get a prime number of grains or rice by that point, so here a lot of
the grains from the 20 have got into our squares here, but let me put these back.
I’ve got 4 on the third square, so let’s look at the third square, how many grains of
rice have I got up to the third square? I’ve got 1 on the first square, 2 on the next
and 4 on the third, 1 plus 2 plus 4 is seven, it’s a prime number. Now Marin
Mersenne thought he was onto something, he moved to the fifth square, another
prime number square and he counted up the number of grains of rice up to the
fifth square and he got 31, another prime. So he began to wonder, maybe this is a
good way to find big primes. Unfortunately it doesn’t always work, if you go to the
eleventh square on the chess board, then there are in total at the eleventh, 2,047
grains of rice up to here. Sounds prime, but actually it’s 23 times 89. So it doesn’t
always give you a prime, but this trick of Marin Mersenne’s, by looking at prime
number squares, is actually responsible for some of the record-breaking primes in
history. So in the reign of Queen Elizabeth I, the record breaking prime was
actually on the nineteenth square, somewhere buried inside here, nineteenth
square. If you count the number of grains of rice up to the nineteenth square, you
will get a prime number, and it was the record prime in the reign of Queen
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Elizabeth. So I would like you to welcome Recreativo Hackney’s sixteenth-century
signing for Queen Elizabeth I, thank you very much. Queen Elizabeth I, now we’re
going to make a space for you, you’re meant to be on the nineteenth square, so
you’ve got to get in there. If you could stand on the nineteenth square. Ok, by the
nineteenth square, how many grains of rice? If you’d like to stretch your number
out so we can see what it is, so let’s turn around so you can show all the audience,
we’ve got 524,287, record prime in the reign of Queen Elizabeth I. Now by the time
Horatio Nelson was fighting the battle of Trafalgar, the record prime had got up to
the thirty-first square in the chess board. So I’d like you to give a big welcome to
Recreativo Hackney’s nineteenth-century signing, Horatio Nelson. Come on down.
Stand on the thirty-first, so Horatio Nelson’s prime, if you’d like to stretch it out, is
a ten-digit prime, and this is the record in the nineteenth century. Now what is
today’s record big prime? Well, which square of the chess board will I have to
move to then? I’ll have to move to the 32,582,657th square, it would be a huge
chess board to try and calculate that one, and I’m going to need a very big football
shirt, I think, for this number, so here’s Recreativo’s most recent signing, and I
want two volunteers to try and put this on because it’s so big. I’m going to have
you two here, we need two next to each other, ok, now we’re going to have a little
team photo here, so if you’d like to move in here, and if you’d like to try and put it
on, so it’s got two heads because it’s so huge. So if you’d try and put that one on,
I’ve chosen one very tall and one very small, so that’s going to be even funnier,
but this prime was discovered just a few months ago by an amateur with his
desktop computer, now it has 9.8 million digits. I mean that’s far too many digits
to put on the front of this shirt, in fact if I tried to read this number out aloud, it
would take me probably a month and a half to say it out aloud, so you’ll be glad to
hear I’m not going to try and do that. But, Marin Mersenne had a formula to
calculate the number of grains of rice on the chess board, and if Queen Elizabeth I
you’d like to turn around, you can show us the formula on the back of your shirt.
Here’s a quick way to find the number, you double 2 nineteen times, take 2 to the
power 19 and then take 1 off, that’s the number of grains of rice on the nineteenth
square. Horatio Nelson if you’d like to turn round, on the thirty-first square the
number of grains of rice is 2 to the power of 31, multiply 2 together 31 times and
take 1 off. So how many is the record prime today? If you’d like to turn round,
that’s good team work, they knew which way to go, I’ll sign them for Recreativo.
Ok what’s the number? You have to double up 32 and half a million times and then
take 1 off to get a prime number, so that’s a huge prime, in fact if you think about
that, Mr Low doubled his noodle 7 times; if he doubled it 90 times, the length of
that noodle would stretch from one side of the observable universe to the other.
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So this noodle, how long would this be? It’s just unimaginably large, we’re
doubling a noodle 32 and half million times, it’s going to be wrapped round the
universe and back again many, many times. Ok, let’s give our signings to
Recreativo Hackney a big round of applause. You see, I think that’s real team
work, they’re trying to get; actually he’s trying to get out of his shirt. I think he’s
celebrating a goal over there so, great. Well, this is the best formula we have for
finding record-breaking primes.
Now if you have a computer at home you can join in the search for these big
primes, and in fact there is a prize of US$100,000 for the first person who can find
a 10 million digit prime. So if you want to join in this cutting-edge research, check
out our website at the end of the programme. But are we sure there’s going to be
such a big prime? Well actually yes, thanks to the Ancient Greeks, who proved
2000 years ago that the primes go on forever, there are infinitely many primes. So,
if Recreativo Hackney want to get infinitely large, we’ll always have a prime
number shirt for them to play in. Now the mystery is, how to find these primes,
and that is the million-dollar problem, solving the mystery of these never-ending
numbers. There are literally thousands of mathematical theorems that are
depending on understanding these most important numbers. But the primes seem
as randomly scattered as the stars in the night sky, there is a million dollars
waiting for the person who can understand how these infinitely many primes are
laid out through the universe of numbers.
Come back tomorrow and you can find out how to win our second million dollar
prize which is for understanding four dimensional footballs. Till then I’m off to
crack the mystery of the never-ending primes. Goodnight.
© Royal Institution and Five 2007
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