`Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?'
These were the opening words of Professor David Hilbert at an international congress of mathematicians in Paris in 1900 during the course of which he outlined a list of the 23 challenges facing mathematics at that time, and threw down the gauntlet to the mathematical community to come up with solutions.
"Who of us would not like to be a millionaire?" would have been an appropriate opening at another, more recent conference in the College de France at which $1 million was offered to anybody who could come up with a solution to any one of seven similar problems over the coming century.
The prize money is being put up by the Massachusetts based Clay Mathematics Institute (CMI). The choice of Paris as the venue for the announcement was no accident, and it is clear that the CMI sees these seven intractables as setting the agenda for this century much as Hilbert's 23 did for the last.
Perhaps the most famous item on the Hilbert list was Fermat's Last Theorem. This was not in fact a theorem, but a conjecture -that there do not exist positive integer solutions of the equation x(to the power of n) + y(to the power of n)for n > 2.
Most people will be familiar with magic sets of integers like 3, 4 and 5 that satisfy the Pythagorean condition 3 (to the power of 2) + 4(to the power of 2)= 5(to the power of 2). But was it possible to find numbers that would satisfy that format to the power three and higher? The answer, it turns out, is no, but it was not until 1998 that a proof of Fermat's conjecture was provided by British mathematician Andrew Wiles.
All but one of the problems from the original Hilbert list have been solved or satisfactorily resolved. The one remaining Hilbert problem is the proof of the Riemann Hypothesis and this has been carried forward to the new Clay list.
The Riemann Hypothesis concerns itself with the distribution of prime numbers. Riemann noted the occurrence of primes was closely related to the solutions of an equation called the Riemann Zeta equation. The first one and a half billion solutions have already been checked, but proof it is true for all solutions would cast some interesting light on the distribution of primes.
A satisfactory solution to this problem would have implications for computer security, as the current cryptographic techniques rely on the difficulty in finding prime factors of very large numbers.
Of the six Clay list newcomers, an interesting problem is the so called `P versus NP problem'. This concerns a comparison between decision and search. Suppose you are at a rugby match and somebody points out a player and asks: Is that Mick Galwey? Almost instantaneously you can confirm or deny as the case may be. On the other hand, if you arrived at the stadium, and someone asked you to point out Mick Galwey, you would have to scan all the players until you recognized the person in question.
Humans can actually cope with this problem far better than machines, and it is for computing purposes that an answer to this question is required. Computers would operate much faster if they only had to decide particular questions rather than having to examine every possibility methodically before making such a decision.
Two of the remaining five concern themselves with topology - the Hodge and Poincar conjectures. The latter has been proved for every number of dimensions except three, which happens to be the accepted number of dimensions in which we live (excluding time).
The Navier-Stokes equations concern themselves with the turbulence and shock waves that follow in the wake of boats and jet aircraft. Nobody has successfully solved these equations without the use of computer-based approximation techniques.
So how do you claim your prize? Well first you must solve the problem to your own satisfaction, and then have your solution published in a recognised mathematical journal. After a period of two years broad acceptance by the mathematical community, the Clay committee will write you a cheque.
Fintan Gibney is an IT consultant with the Irish software company SmartForce.
You can visit a website to learn more about this maths competitionwww.claymath.org/ prize-problems/index.htm