## The Prime Numbers Have It

Évariste Galois: Died in a duel at the age of 20 but left a lasting legacy

In June the first Breakthrough Prizes in mathematics were announced: \$3 million for each of five recipients, three in America, one in France, and one who divides his time between Britain and America. The new prizes, founded by internet moguls Mark Zuckerberg and Yuri Milner, were announced last December at a ceremony to award similar prizes in physics and the life sciences. Narrowly focused on particular specialisations, these differ from the mathmatics prize, which is spread across the whole subject.

The big difference between mathematics and other sciences is that maths has no need for expensive experimental work, and mathematicians are free to follow their own instincts-trying to propel them in a particular direction would be rather like herding cats. With no need for experimental corroboration and inspiration, mathematics is very much an exercise of the mind, success residing in the proving of theorems.

A theorem might start out as a well-known conjecture — the 1859 Riemann Hypothesis with its implications for the distribution of prime numbers is the most famous — but it can also be far more mysterious, such as an unlooked for connection between two murky areas of mathematics. A great example of this occurred in 1832, when a 20-year-old French mathematician Évariste Galois sat down to write a letter outlining his mathematical results. The next morning he died in a duel. What Galois had done was connect the problem of solving algebraic equations to the study of symmetry. The more symmetries there are among the solutions to an equation, the tighter their bonding and the harder it is to extract them by means of a formula. He gave a precise connection between equations and symmetry, and had this been a known conjecture his work would have received immediate scrutiny. Prizes would have been awarded — though of course he was dead by then — but it was completely unexpected. Reactions only came years later and his fame was entirely posthumous.

Galois’s prominence as a political revolutionary reputedly brought 300 people to his funeral, but Zuckerberg and Milner want mathematicians to be well-known for their mathematical prowess. Milner, who quit the world of physics and made a billion dollars in business, says, “Scientists should be better appreciated. They should be modern celebrities, alongside actors and entertainers. We want young people to get more excited.”

It’s a fine idea, though celebrity status does not suit all mathematicians. Apparently when Andrew Wiles proved Fermat’s Last Theorem, a clothing company wanted him to model a pair of jeans — he wasn’t interested. And when the Russian mathematician Grigori Perelman won mathematics’ highest honour for his proof of the Poincaré Conjecture (about three-dimensional curved spaces) he refused to attend the ceremony or accept the prize money.

Of course the settling of an old conjecture can readily be publicised, but what of the deeper undercurrents that barely cause a ripple on the public surface of mathematics? Unfathomable, even to the educated layman, these are where the new ideas lie. Work on an old conjecture that does not delve into such depths usually results in the efforts being stillborn. A lovely fictional example appears in Uncle Petros and Goldbach’s Conjecture. In Apostolos Doxiadis’s novel, the protagonist gives up a very promising mathematics career to concentrate all his efforts on the conjecture that every even number is the sum of two prime numbers, but in a William Golding-like descent into madness, he fails.

Problems about prime numbers can indeed drive you crazy. Some are easily settled, like the fact, proved by Euclid 2,300 years ago, that there are infinitely many primes, but consider a similar question that is not so easy. Take a sequence of numbers such as 3, 10, 17, 24, 31, where the difference between one and the next is constant. This is called an arithmetic progression. If there is no common factor, must such a sequence contain infinitely many primes? At one time this could only be proved in special cases, but in 1837 a young German mathematician, Peter Gustav Lejeune Dirichlet, provided a universal proof, revealing techniques previously unknown, which in turn led to new questions and results.

Switch the problem around and consider arithmetic progressions within the infinite set of prime numbers. Can they have any desired length? For instance 5, 11, 17, 23, 29 has length 5, but any length is not so easy. It was proved in 2004 in joint work by Ben Green in Oxford and Terence Tao in Los Angeles.

Slowly but surely these problems are cracked, though it may take hundreds of years. Here is one that hasn’t got there yet: are there infinitely many pairs of prime numbers differing by only 2? No one knows, but last year a Chinese-American mathematician, Yitang Zhang, proved that if the difference was at most 70 million there were infinitely many such pairs. This put the cat among the pigeons, and Terence Tao started a project to prove the same result with a smaller difference. His project, involving a more general claim — so often the case in mathematics — had the difference down to 246 in April, but as Tao writes, “[This] is a nice place at which to ‘declare victory’. . . it may be smarter to actually let things rest for a while in case some external development makes further progress a lot easier.”

Tao is one of the five recipients of the new Breakthrough Prize in mathematics, along with Simon Donaldson (Imperial and Stony Brook), Maxim Kontsevich (Institut des Hautes Études Scientifiques), Jacob Lurie (Harvard) and Richard Taylor (Institute for Advanced Study, Princeton). The external developments he alludes to are the key to serious progress, and those who make such strides are candidates for the most prestigious prize in mathematics — the Fields Medal of which Tao himself was a recipient in 2006.

Awarded once every four years to four people, at most, the Fields Medal has a cachet like that of the Nobel Prize, though at 15,000 Canadian dollars the financial rewards are far lower. Yet unlike the Nobel Prize, often conferred for achievements long past, its aim is to encourage new work, and an unwritten rule says you cannot be over 40 to win one. As a prize for the relatively young, its recipients are usually not well-known but their names are eagerly awaited, and this year for the first time a woman, Maryam Mirzakhani at Stanford, is one of the four winners.

These prizes may not create celebrities, but if they help raise public awareness of mathematics that is enough. One occasionally meets people who think the subject was all finished long ago, yet it constantly renews itself with new ideas, results and methods. Humanity’s desire to plunge ever deeper into numbers and patterns, and climb ever higher to see hitherto undreamed — of connections must never cease.