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The Music of the Primes
Mathematicians and the Pursuit of their Greatest Prize
Prime numbers are the atoms of mathematics. They can’t be broken down into
simpler units – they are, by definition, indivisible by any other number. And they
seem to be fundamental to all numbers: for over 3,000 years, mathematicians have
known that every other whole number can be built by multiplying two primes. Put
simply: primes are to maths what letters are to words, and notes are to music.
So primes are simple: yet they’re also incredibly elusive. As long ago as 300 BC,
Euclid proved that there must be an infinite number of them. But we seem to find
them in the most unexpected places. They appear to be distributed very unevenly.
And no matter how hard they’ve tried, mathematicians have never managed to find a
reliable way to predict where a prime number will appear.
They’ve also discovered that primes often seem to have an important role in the
natural world – and that they can crop up in the most surprising places. For example,
in the forests of North America, there is a species of cicada that depends on prime
numbers for its survival. These cicadas spend most of their existence underground –
but once every 17 years, they surface en masse, and then feast and mate for six
weeks. After their orgy, they die – until, 17 years later, the next generation appears.
The reason for this odd life-cycle, it seems, is because of the breeding patterns of
their main predators, which are at their most prevalent every 5 years. The clever
cicadas have worked out that breeding every 17 years minimises their risk of
encountering their enemies.
As a discipline, mathematics prides itself on precision. To a greater extent than any
other science, maths demands proof: assertions based on hypotheses simply won’t
do. So imagine how enormously frustrating it is for mathematicians to admit that,
despite the determined efforts of some of the greatest mathematical minds, they still
can’t tell with certainty where a prime number will arise.
The hunt for a pattern in the distribution of primes has been an obsession for some of
the giants of maths. Three of the titans of 18th and 19th Century mathematics, Euler,
Jacobi, and Gauss, tried to find one - but failed. Yet mathematicians are all agreed
that the pattern can’t be random: surely, to paraphrase Einstein, God doesn’t play
dice with the universe of primes?
Then, in the middle of the 19th Century there was a vital breakthrough. One of
Gauss’s students, Bernhard Riemann, developed a theory that many mathematicians
believe provides an answer to the conundrum. His theory was based on the wavy
lines that are generated by a mathematical device called the zeta function.
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Riemann’s Zeta Landscape
As maths functions go, the zeta function isn’t especially complicated. But if you plot
the zeta function graphically, it looks like a vast undulating landscape, studded with
steeply rising hills and plummeting valleys. At his base in the University of Gottingen
in Germany, Riemann discovered that if you can locate all of the places where the
landscape dips to ‘sea level’, these points can be used to predict the behaviour of the
primes. But rather than being randomly scattered around the landscape, Riemann
noticed a fascinating pattern. The points seem to line up along some mysterious
straight line in the landscape.
Bernhard Riemann
Riemann had discovered, it seemed, the mathematical Holy Grail: a formula to
predict with certainty the distribution of primes.
Well… not quite. Riemann had developed a fascinating hypothesis, by identifying a
function that seemed to have real predictive power - yet he hadn’t provided a rigorous
mathematical proof for any of it.
Riemann is thought to have been closing in on a proof just before his death in 1866.
But we’ll never know how close he was to providing a solution to the most perplexing
problem in mathematics – because shortly after his demise, his housekeeper threw
many of his unpublished manuscripts on the bonfire. Tantalisingly, there is evidence
that a little black book used by Riemann to record his discoveries was among the
papers that escaped the flames. But mysteriously, it has since disappeared from the
archive.
What survives of Riemann’s brilliant work has been a shot from a starting pistol,
launching a race to find the proof that would unlock the strange secret that lies
behind the distribution of the primes.
Since Riemann’s demise, most of the world’s great mathematicians have grappled
with this problem. David Hilbert, Alan Turing and Kurt Godel all tried. Even the
extraordinary maths genius Srinavasa Ramanujan tried. But they all failed.
Ramanujan’s case perhaps best illustrates the difficulty of the problem. By any
measure, Ramanujan was a mathematical phenomenon. A penniless port authority
clerk from the town of Erode in southern India, Ramanujan had no formal
mathematical education. Yet after reading only one basic maths text book, he was
able to reconstruct the entire history of Western maths – entirely on his own.
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Srinivasa Ramanujan
After being plucked from obscurity and penury by the distinguished mathematician
G.F. Hardy, Ramanujan was brought to Cambridge, and there produced a startling
number of original solutions to several ancient problems. But though he developed
some important insights into the fundamental qualities of numbers, even the
extraordinary Ramanujan could make only a limited contribution to the conundrum of
the primes.
But recently, the possibility of a real breakthrough has emerged – from an unlikely
quarter. Research by quantum physicists has shown unexpected connections
between the frequencies of Riemann's ‘music’ and the energy levels found in the
nuclei of Uranium atoms. Riemann’s points and these subatomic quantum effects
seem to share very similar and distinctive patterns. This surprise discovery was made
during tea at the Institute at Princeton when a mathematician and a physicist were
idly chatting. Perhaps quantum physics, rather than pure mathematics, will ultimately
provide the answer to understanding the primes.
Grasping the true nature of prime distribution remains elusive – but perversely,
identifying primes that were once unimaginably huge is now routine stuff. By using
supercomputers, and working collaboratively via the internet, prime hunters regularly
discover some truly massive primes. The largest yet unearthed, in February 2005, is
225,964,951-1 - a number that is over 7.8 MILLION digits long.
Yet frustratingly, a definitive proof still escapes the mathematical world. Perhaps this
is just as well. If someone did produce a formula that would enable us to predict the
distribution of primes, it would bring them global fame - and a $1 million prize. But it
would also bring down the banking system as we know it, since all existing internet
banking security systems are based on the unpredictability of prime distribution.
The beguiling story of the search for a pattern in the primes is revealed in Marcus du
Sautoy’s The Music of the Primes – a chronicle of the long struggle to find a solution
to the most fundamental and frustrating puzzle in all mathematics. As a follow-up to
his recent appearances on the BBC FOUR series Mindgames, we want du Sautoy to
reveal the remarkable story of this epic pursuit – a 3,000 year-long story of
mathematical joy and despair, frustration and insight, blinding light and blind alleys.
It’s a journey that will take du Sautoy from ancient seats of learning in Greece, to the
modern academic campuses of Gottingen, Princeton, and Cambridge. On the way,
he’ll encounter swarms of cicadas in North America, the surviving relatives of
Ramanujan in southern India, quantum physicists working with the huge atomsmashers at CERN in Switzerland, and two extraordinary autistic patients of the
psychiatrist Oliver Sacks – twins, whose only form of communication involves the
exchange increasingly large prime numbers.
Like du Sautoy’s book, this film will describe this great intellectual enterprise in an
intelligent yet thoroughly accessible way. He’ll couple the powerful central narrative
of this great intellectual battle with compelling visual metaphors, and will relate key
episodes in the lives of the great mathematicians who’ve sought the answer to the
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problem of the primes. In a film richly illustrated by rare archive footage, and original
location filming, we’ll use the latest computer-generated imaging technology to
produce an ambitious, spectacular three-dimensional graphical representation of
Riemann’s zeta landscape, to map the ancient pursuit of the most precious prize in
mathematics.
David Okuefuna
BBC Arts
March 2005
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