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### Documentary Description

Prime numbers - those figures which refuse to be divided neatly by anything other than one and themselves - are surprisingly stubborn and persistent beasts. Join Marcus du Sautoy as he explores their strange behaviours, and offers his personal welcome.

Prime numbers have fascinated people for centuries and I’m no exception. I live in a prime obsessed world. I have a prime number of children – twin girls - and a boy makes three. My house number is prime, 53; my car registration is prime, 47; my car is a Primera!

I even persuaded the football team I play for to wear prime number shirts. So we do, from 2 up to 41. And, ever since, the results have been remarkable. We’ve shot to the top of the league.

But my other passion is the search for patterns and the primes offer the ultimate challenge. These fundamental numbers look as wild as stars in the night sky. How did Nature choose the primes?

This is the story of a 2000-year journey across cultures and continents in humankind’s attempts to solve the riddle of these enigmatic numbers. Welcome to The Music of the Primes.

About the Programme

For more than 2000 years, a mathematical riddle has baffled the world’s greatest minds. It’s a problem of such difficulty that it has tormented those mathematicians brave enough to tackle it. Some have given up in despair. Others have been driven mad.

Yet it’s a conundrum that helped Britain to win the Second World War; that was instrumental in the birth of the computer; and that has shed light on the behaviour of atoms, the building blocks of matter itself.

Today, the on-line financial world depends upon its impenetrability. If a solution to it were to be found, it could bring the financial world to its knees. It’s hardly surprising that a prize of For more than 2000 years, a mathematical riddle has baffled the world’s greatest minds. It’s a problem of such difficulty that it has tormented those mathematicians brave enough to tackle it. Some have given up in despair. Others have been driven mad. ,000,000 has been offered to whoever cracks it.

The mystery that has confounded mathematicians for centuries is the riddle that surrounds the distribution of prime numbers. Primes are fundamental to mathematics: they are, after all, the basic building blocks from which all other numbers can be built. Yet they seem to surface entirely randomly along the number line. But are the primes truly random – or is there some hidden pattern? It’s the greatest unsolved problem of mathematics - and whoever cracks it will achieve mathematical immortality.

In The Music of the Primes, Marcus du Sautoy investigates the fascinating story of the great mathematicians – including Carl Friedrich Gauss, Bernhard Riemann, G.H. Hardy, Srinivasa Ramanujan, and Alan Turing - who’ve grappled with the problem of the primes.

Filmed on location in Princeton, Las Vegas, Athens, Madras, London, Cambridge, and Gottingen in Germany, Marcus talks to some of the world’s leading mathematicians who’ve been listening to The Music of the Primes.

Marcus du Sautoy is a professor of mathematics at All Souls' Oxford. He is the author of the book The Music of The Primes, upon which this series is based.

Director: Robin Dashwood

Executive Producer: David Okuefuna

A Prime Primer

They're astonishingly powerful numbers. Simon Singh introduces the primes.

Mathematicians love prime numbers in much the same way that chemists love atoms and biologists love genes. Just as atoms are the building blocks of the matter around us and genes are the building blocks of life, prime numbers are the building blocks of mathematics.

Hence, the study of prime numbers has had a tremendous impact on everything from illuminating unexplored areas of pure mathematics to inspiring new technologies for protecting our most secret communications.

Prime numbers, for those who have forgotten, are those numbers that cannot be divided by any other number except 1 and itself. So 5 is prime because nothing will divide into it, but 6 is not prime because it can be divided by 2 and 3. All numbers can be broken down into a series of one or more primes multiplied together, so the prime 5 is just 5, while 6 is 2×3, and a more complex number such as 90 is 2×3×3×5. Because all numbers can be broken down into primes, mathematicians believe that studying prime numbers will lead to an understanding of all numbers.

Mathematicians have been pondering primes for thousands of years, and one of the first major breakthroughs was made by Euclid in around 300 BC in Alexandria. He asked himself the question, do the primes eventually run out or is there an infinity of primes? Is there a biggest prime, or can I always find a bigger one? It might seem that questions about infinity require ridiculously complicated solutions, but Euclid solved this problem with a simple and elegant argument, which showed that there is, indeed, an infinite number of primes.

Euclid’s argument started by assuming the opposite, namely that there is a finite number of primes. Let’s imagine that someone claims that 2, 3 and 5 are the only primes in the world. However, if we multiply them together (2×3×5) and add 1 we get 31. Clearly 2, 3 and 5 will not go into 31, so either there are a couple of primes that we do not know about or 31 itself is a new prime, and indeed the latter is the case.

So our list of primes is longer (2, 3, 5 and 31), but still we know that our list of primes is incomplete, because if we multiply 2, 3, 5 and 31 and add 1 then we prove again that our list of primes is incomplete. In short, whatever list of primes someone generates, it is always possible to prove that there are more primes, and therefore the complete list of primes must be infinitely long.

However, there are many questions about prime numbers that continue to baffle mathematicians. In particular, nobody has worked out why the prime numbers are distributed in such a peculiar way.

In the 100 numbers before 10 million there are ten primes, whereas in the 100 numbers after 10 million there are only two. Sometimes we encounter neighbouring primes (so-called twin primes), separated only by an even number, but they might be followed by an enormous dearth of primes. The primes spring up unpredictably, like weeds among the other numbers. The Riemann Hypothesis is a 145 year-old conjecture that would give a great insight into this sporadic distribution of primes, but as yet it remains unconquered, making it a mathematical Everest still in search of its Hillary.

Euclid’s proof of the infinity of primes and the Riemann Hypothesis both touch on areas of pure mathematics, but prime numbers are also at the heart of some very applied maths. Most importantly of all, the study of prime numbers has transformed personal privacy and global economics. This is because prime numbers form the basis of many modern encryption algorithms, in particular a technology known as public key cryptography.

This system overcomes an age-old problem in communication, known as the key distribution problem. This means that a sender (Alice) who encrypts a message and sends it to a receiver (Bob) also has to somehow give him the key, the secret piece of information that allows him to decrypt the message. An analogy is to imagine that Alice wants to send Bob a precious jewel, so she puts it in a box and locks it with a key. But when the box is delivered to Bob, he cannot open it because he does not have the key. In the past, Alice would have been forced to meet Bob beforehand to give him a copy of the key or use a secure courier to deliver the key separately.

However, public key cryptography avoids any need for key distribution. This time when Alice wants to send a jewel to Bob, the revised analogy involves padlocks. Bob starts by sending Alice an open padlock, while he keeps hold of the key. Alice would then use the padlock to lock the jewel in a box, because she does not need the key to snap the padlock shut. Finally, when Bob receives the box, he can open the padlock, because he retained the key. This is the perfect security system – no key was ever distributed, the box was secure in transit, and yet Bob could open it.

Encryption based on this sort of technique has enabled everything from e-commerce to secure emails, from pay-TV to private cell phone communication. Clearly real padlocks are not used, but instead we use mathematical padlocks that rely on the multiplication of prime numbers. It is very simple to multiply two prime numbers to obtain their product (11×13=?), but it is very hard to examine the product of prime numbers and deduce the two prime factors (?×?=437). This is rather like a padlock, because it is one-way function. Padlocks are easy to lock, but hard to unlock, while primes are easy to multiply, but hard to factor. There is more to public key cryptography than primes, but they certainly provide the foundation of the system.

The difficult of factoring a product into its prime components increases rapidly with the size of the numbers involved. If you are sceptical that factoring is really so difficult, then you might want to try the challenge offered by the American security company RSA. They offer $100,000 for anybody who can factor this number:

13506641086599522334960321

62788059699388814756056670

27524485143851526510604859

53383394028715057190944179

82072821644715513736804197

03964191743046496589274256

239341020864383202110372958

725762358509643110564073501

508187510676594629205563685

529475213500852879416377328

533906109750544334999811150

056977236890927563

Alternatively, you can earn $100,000 by being the first person to discover a prime with over 10 million digits. The prize is offered by the Electronic Frontier Foundation, and your best chance of winning the money is by joining the Great Internet Mersenne Prime Search (GIMPS), which is a co-ordinated attempt involving thousands of amateurs united by the Internet.

However, the biggest prize in prime numbers was put up by the Clay Foundation. They are offering a $1 million reward for anyone who can prove the notorious Riemann Hypothesis. Good luck!

Sources: BBC