$7m reward to solve maths puzzles

Prizes totalling $7m (£4.2m) were offered yesterday for solutions to the seven most important unanswered questions in mathematics.

Prizes totalling $7m (£4.2m) were offered yesterday for solutions to the seven most important unanswered questions in mathematics.

At a Collÿge de France conference in Paris, the Clay Mathematics Institute of Cambridge, Massachusetts, offered $1m each for solutions to the problems which defied the combined number-crunching brain-power of the 20th century.

A century ago, a German mathematician, David Hilbert, startled a conference in Paris by setting out 23 mathematical problems that had defeated the 19th century. Most of the problems have been solved but others have emerged. The struggle to find the answers has helped to shape scientific developments of the past 100 years, from computers to nuclear fission. The Clay Institute, created by the US businessman Landon Clay, believes the answers to its seven problems (which include only one of the old ones) could open doors to more scientific discoveries.

Although they are all "pure" mathematical problems, with no evident technical applications, the solutions could uncover areas for scientific inquiry. The prize offer remains open indefinitely.

"In every discipline in mathematics, there is a Mount Everest, which has never been climbed," said Professor Alain Connes. "We know that by climbing to the top of that mountain, we will have a fantastic view and we shall understand a lot of things which are now out of reach."

The seven problems are "P versus NP"; the Riemann Hypothesis; the Poincaré Conjecture; the Hodge Conjecture; the Birch and Swinnerton-Dyer Conjecture; the Navier-Stokes Equations; and the Yang-Mills Theory.

Riemann's Hypothesis is the only problem left from the 1900 list, now regarded as the most important remaining pure problem in pure mathematics. Here is the Clay description: "Some numbers have a special property that cannot be expressed as the product of two smaller numbers, eg 2,3,5,7 etc. Such numbers are called prime numbers, and they play an important role, in pure mathematics and its applications.

"The distribution of such prime numbers among all natural numbers does not follow any regular pattern, but the German mathematician GFB Riemann [1826-66] observed that the frequency of prime numbers is very closely related to the behaviour of an elaborate function 'Zeta (s)' called the Riemann Zeta function.

"The celebrated Riemann hypothesis asserts that all interesting solutions of the equation Zeta (s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers."

Readers are advised to start now. Write on only one side of the paper. You have as many decades as you need.