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.