The Music of the Primes, by Marcus du Sautoy

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...
Prime numbers are the atoms of arithmetic. At school, we are taught that they are divisible only by themselves and the number one. What we are rarely taught is that they represent one of the most tantalizing enigmas in the pursuit of human knowledge. How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks.

Prime numbers are the most important numbers in mathematics. Every number is built by multiplying together these indivisible numbers. For example 105 is built by multiplying the primes 3, 5 and 7. The primes are for the mathematician what atoms are for the chemist. They are the hydrogen, helium and lithium of the world of numbers. Chemistry has been very successful at producing a list of all the atoms that exist in Nature. Called the Periodic Table, it lists 109 chemical elements from which all molecules can be built.

Despite several millennia of investigation mathematicians are still struggling to understand their own mathematical Periodic Table of primes. The reason they are so hard to understand is that, unlike atoms, the primes go on for ever. In the first great Theorem of mathematics, the Greek mathematician Euclid proved that there are infinitely many prime numbers. Gone is the hope of just making a list of 109 primes, like the chemists, from which all numbers can be built.

Understanding the primes goes to the heart of what it means to be a mathematician. Mathematics is the science of patterns. Most people have the impression that mathematical research consists of doing long division to a lot of decimal places. But this is as crazy as thinking that professional musicians just play their scales faster and higher than anyone else. Mathematicians spend their lives striving to look for patterns, for logic and order in the chaos that surrounds us. Faced with the primes stretching out to infinity, mathematicians are hunting for some structure and pattern which can help us understand how Nature chose these elusive numbers.

The trouble is that as one looks through a list of primes it is very hard to predict when the next prime will appear. The numbers look more like lottery ticket numbers than numbers with some pattern. As one stares out to the infinite depths of the universe of numbers, the primes look as random as the stars in the night sky.

What makes a good mathematician is someone who can think laterally, who can look at the problem in a new way. In the nineteenth century a German mathematician called Bernhard Riemann found a new perspective on the primes which he believed would explain how the primes are laid out amongst all the numbers. Like the cosmologists who discovered the source of the stars in the night sky, Riemann uncovered what he believed was the Big Bang behind the primes.

Looking at the primes' apparent chaos from a fresh perspective, the German mathematician got an inkling of a subtle underlying harmony - the music of the primes. On the strength of his expert intuition, he made a bold prediction about the primes' hidden music, later dubbed the Riemann Hypothesis. Sadly, no one knows if Riemann actually found the key to the puzzle of prime numbers. He died before providing the proof, and his housekeeper burned many of his personal papers.

Since his death, scores of mathematicians, plus more than a few physicists and high-tech pioneers, have set out to conquer the Riemann Hypothesis. Today, the victor stands to gain one million dollars - courtesy of a Boston businessman - along with the glory. The story of humankind's attempts to master the primes reads like an unsolved murder mystery. It is a story of strange journeys, last minute escapes from death and the unquenchable thirst for knowledge. Its dramatis personae include a French spy, a Hungarian Jew fleeing Nazi persecution, a clerk from India, and John Nash (played by Russell Crowe in A Beautiful Mind) - whose "prime obsession" may have contributed to his mental breakdown.

Mathematicians study the primes because of their beauty and universal character, like a common language of the cosmos. But today there are very commercial reasons for studying the primes - they are the key to modern codes. Every time someone sends their credit card number securely across the Internet, they are using properties of the primes to keep that number secret. To crack codes on the Internet, a hacker has to find a way to discover the primes that built an e-business's code number. It's like the chemist who wants to know what atoms make up salt. Chemistry has successfully invented a machine called a spectrometer which takes the salt and tells the chemist that Sodium and Chlorine are the atoms which built this molecule.

But the mathematicians have so far failed to discover a fancy prime number spectrometer which can quickly take a large number and tell you the primes that built that number. Understanding the primes holds the prospect not only to unlock the greatest mathematical enigmas of all time but could bring e-business crashing to its knees. Mathematics is still waiting for the person who can make that elusive breakthrough and make the primes sing.

JOC/EFR November 2017

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