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# Lecture 1: The curious incident of the never-ending numbers

Did you know how many different types of symbols have been used to represent numerals across the world over time? Lots, including Babylonian, Egyptian, Mayan, Roman and Chinese; but the ones most commonly used now are Arabic numerals. Of all numbers we use, did you know that the primes are special? A prime is a number that has exactly two divisors: 1 and the number itself. In this lecture Marcus helps us discover the different ways that people through history have tried to find out whether a number is a prime. But no-one can claim to have collected all the primes, and here Marcus provides the proof of how we can be sure that primes are infinite.

# Further and Deeper

## Try this out

The Mersenne mystery (PDF Document 292KB - new window)

## More Maths

The prime numbers carry on forever, but how do we know? Download this pdf to see why we can prove that the list of primes is never-ending, and what makes this proof so special.

Infinite primes (PDF Document 523KB - new window)

## Lecture 1 transcript

The curious incident of the never-ending numbers (PDF Document 156KB - new window)

View a list of the first 10,000 primes. Can you spot the pattern?

The first 10,000 primes (external link - new window)

Explore the website that accompanies Professor Marcus du Sautoy's book: The Music of the Primes.

The music of the primes (external link - new window)

It's easy to see that 105 is 5 x 3 x 7, but for much larger numbers it's not easy to break a number into its prime divisors. This link takes you to a number so difficult to divide that there is a US\$200,000 reward for doing it!

Prize primes (external link - new window)

## Video clip - Unexploded primes

Marcus demonstrates a quick way to find prime numbers.

(Press play to start the video.)

This exciting, interactive site has been designed using Flash 8. In order to get the full experience, you will need to download Adobe Flash version 8.

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.

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).

OK, we move along. Next, we move along here we find number 2. OK, so that's the first prime number. So I'm going to try and knock out all the balloons which are divisible by two. Anything in the 2 times table isn't a prime, so I'm going to knock out all the even numbers, my first volunteer here is going to help me do it. So what's your name sir?

Robert.

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. We're going to count him down, we're going to go three, two, one, go. There we go, fantastic; we've knocked out half the balloons there. Oops, we've 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's going to knock out everything in the 3 times table. Let's count her down, three, two one, go. OK, not bad, 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, 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 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, that 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.

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.

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).

OK, we move along. Next, we move along here we find number 2. OK, so that's the first prime number. So I'm going to try and knock out all the balloons which are divisible by two. Anything in the 2 times table isn't a prime, so I'm going to knock out all the even numbers, my first volunteer here is going to help me do it. So what's your name sir?

Robert.

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. We're going to count him down, we're going to go three, two, one, go. There we go, fantastic; we've knocked out half the balloons there. Oops, we've 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's going to knock out everything in the 3 times table. Let's count her down, three, two one, go. OK, not bad, 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, 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 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, that 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.