A subject in the prime of life

The end of maths is greatly exaggerated and there are still many hypotheses to prove, argues Ian Stewart.

One measure of the health of today's mathematics is the rate at which the great unsolved puzzles are being disposed of. The latest is Fermat's Last Theorem, a deceptively simple conjecture made by Pierre de Fermat, a lawyer with a mathematician's soul. It has taken 350 years to prove Fermat was right: Andre Wiles of Princeton University made the final breakthrough in October 1994.

The Last Theorem - so named because it was the only one of many statements made by Fermat that had not been proved by his successors - states that two perfect cubes cannot add up to a perfect cube, and that the same goes for all higher powers. Its proof ranges over some of the deepest and most difficult areas of modern mathematics, and is accessible only to experts. It is not the only big unsolved problem to have cracked wide open under the onslaught of modern methods. Another is the Four Colour Theorem, which states that any map in the plane can be coloured with four colours, in such a manner that no two adjacent regions are given the same colour. That one took some clever ideas about plane geometry and two thousand hours of computer time. Given these successes, it would be tempting to conclude that mathematicians are rapidly polishing off all the problems of their subject, so that we will shortly reach the End of Mathematics.

Not so. There are plenty of big problems left for future generations to cut their teeth on, and new ones arrive on the scene every day.

So: what are the big unsolved questions? There are several about prime numbers - which, you will recall, are numbers like 5, 7, and 19, which are not divisible by any smaller number except 1. In 1742 an amateur called Christian Goldbach wrote to Leonhard Euler, the most prolific mathematician of all time, observing that every even number appeared to be the sum of two primes. For example 14 = 3+11. Euler was unable to verify Goldbach's observation in general, and nobody else has been able to either. However, there has been quite a bit of progress. In 1931 L. Schnirelmann proved that every even number is the sum of at most 300,000 primes; by 1937 I. M. Vinogradov had improved that to four primes - but perhaps with finitely many exceptions. How big those putative exceptions might be, he did not say; but in 1956 K. G. Borodzkin showed that they have at most four million decimal digits. The best result so far was proved in 1966 by Chen Jing Run: every sufficiently large even number is either the sum of two primes, or the sum of a prime and a product of two primes.

Topology - often referred to colloquially as "rubber sheet geometry", because it permits continuous distortions of shapes - is notorious for open questions. But the most famous is the Poincare Conjecture of 1904. Actually Henri Poincare did not guess at an answer, saying only that "this question would lead us too far astray", so technically the word "conjecture" is incorrect; but that is what everybody calls it. As a warm-up, observe that the surface of a sphere has two simple properties. First, it is connected: all in one piece. Second, it is simply connected: if you draw a closed loop on a sphere, then you can always shrink that loop down to a point in a continuous manner. Roughly speaking, this says that the sphere has no holes. Topologists could also prove the converse: any connected, simply connected surface is equivalent to a sphere. Poincare needed to know whether the same result is valid for three-dimensional shapes instead of surfaces, but he could not decide the issue. In 1961 Stephen Smale generalised Poincare's question to shapes of seven or more dimensions, proving that there the answer is "yes". Soon after, John Stallings did the same for six dimensions and Christopher Zeeman for five. In 1982 Michael Freedman invented a new approach to four-dimensional shapes and again got an affirmative answer. That left only the three-dimensional version of the question - the one that Poincare had stated in the first place. It remains unanswered to this day. An affirmative answer would sort out most of three-dimensional topology; a negative one would shock the mathematical community rigid.

Probably the oldest unsolved mystery is the Kepler Problem. In 1611 Johannes Kepler, who achieved scientific immortality with his discovery that planetary orbits are ellipses, wrote a small booklet as a new year's gift to his sponsor. It was about snowflakes, and his musings on their hexagonal symmetry led him to anticipate the modern atomic theory of crystals. Along the way he discussed how to pack lots of spheres together, asking (though not in those words) whether the most efficient packing is the one typically used by greengrocers for oranges and apples. Start with a flat layer arranged in a honeycomb pattern like snooker balls at the start of a game; pile another layer on top, resting the new balls on the gaps between those in the layer beneath, and so on. Kepler stated that such a packing "will be the tightest possible", a phrase that has passed into anecdotal history as the statement that "most mathematicians believe, and all physicists know". In 1990 Wu-Yi Hsiang announced a proof that Kepler was right. Unfortunately, the more the experts looked at the proof, the less happy they became, and the problem is still open.

The Holy Grail of mathematical research is the Riemann Hypothesis, stated in 1859 by Bernhard Riemann. Its implications are far-reaching: for example, many suspected properties of prime numbers hinge upon it. Its origins go back to two of the greatest mathematicians of all time, Carl Friedrich Gauss and Euler. Gauss observed "experimentally" that the number of primes up to some given number is approximately equal to that number divided by its logarithm, and Euler discovered some curious formulas relating prime numbers to what is now known as the "zeta function". The zeta function of 2, for example, is the infinite sum 1 + 1/4 + 1/9 + 1/6 + I where now the numbers are the cubes. In general the zeta function of a number n is the sum of the reciprocals of the nth powers. In 1896 Jacques Hadamard used the zeta function to prove Gauss's conjecture about primes, opening up a huge area known as "analytic number theory", which uses calculus to study whole numbers. Riemann's contribution was to define the zeta function of any complex number whatsoever, and to relate it to all sorts of questions about the size and form of primes. Because of the way complex analysis works, a key question is to determine the "zeros" of the zeta function - the numbers whose zeta function equals zero. Some zeros are easily found, namely -2, -4, -6, and so on. Riemann noticed that all of the other zeros seemed to have a special form - their "real part" was 1/2 - and his hypothesis is that there are no exceptions to this rule. We know there are none among the first one and a half billion zeros - but for all anyone can tell, the very next one might have the wrong real part, so that is not a proof. A proof - or disproof - might come this year, in a thousand years' time, or never.

Those are a few of the old questions. The new ones would fill many books, for as mathematics grows, so does its unexplored boundary. The End of Mathematics? Rubbish. We have hardly got started.

Ian Stewart is professor of mathematics at Warwick University. A new edition of the classic What Is Mathematics? by Richard Courant and Herbert Robbins, revised and updated by Ian Stewart, is published by Oxford University Press.

SUMS STILL TO SOLVE ON A WET AFTERNOON

* The Poincare Conjecture * Goldbach's Primes * The Kepler Problem * The Riemann Hypothesis