Some mathematical problems can take a while to figure out. Take Fermat’s last theorem for example which was first proposed by Pierre de Fermat in 1637 and managed to vex mathematicians the world over until English mathematician Andrew Wiles managed to prove it in 1995, some 358 years later. There’s quite a few problems like that in mathematics that have stood a long time without resolution in part because the math involved is ridiculously complex. Two other such problems are The Riemann hypothesis, which deals with prime numbers and was first proposed in 1859, and The Poincaré conjecture, first proposed in 1904 that deals with the shapes of spaces (topology). These two problems are included on the Clay Mathematics Institutes’s list of seven “millennium problems” some four years ago that they were offering $1 million apiece to whomever could solve them. Word has it that both may be close to being proven which could have a major impact on a couple of different things if the proofs hold up.
The Riemann hypothesis has the biggest potential economic impact if it has been solved because it would explain the apparently random pattern of prime numbers which are an integral part of the current methods of internet cryptography used to keep your credit card information safe when doing online transactions:
This year Louis de Branges, a French-born mathematician now at Purdue University in the US, claimed a proof of the Riemann hypothesis. So far, his colleagues are not convinced. They were not convinced, years ago, when de Branges produced an answer to another famous mathematical challenge, but in time they accepted his reasoning. This time, the mathematical community remains even more sceptical.
“The proof he has announced is rather incomprehensible. Now mathematicians are less sure that the million has been won,” Prof du Sautoy said.
“The whole of e-commerce depends on prime numbers. I have described the primes as atoms: what mathematicians are missing is a kind of mathematical prime spectrometer. Chemists have a machine that, if you give it a molecule, will tell you the atoms that it is built from. Mathematicians haven’t invented a mathematical version of this. That is what we are after. If the Riemann hypothesis is true, it won’t produce a prime number spectrometer. But the proof should give us more understanding of how the primes work, and therefore the proof might be translated into something that might produce this prime spectrometer. If it does, it will bring the whole of e-commerce to its knees, overnight. So there are very big implications.”
The Poincaré conjecture also carries with it some major ramifications of it’s been proven, though of a more positive nature:
Bernhard Riemann and other 19th century scholars wrapped up the mathematical problems of two-dimensional surfaces of three dimensional objects – the leather around a football, for instance, or the distortions of a rubber sheet. But Henri Poincaré raised the awkward question of objects with three dimensions, existing in the fourth dimension of time. He had already done groundbreaking work in optics, thermodynamics, celestial mechanics, quantum theory and even special relativity and he almost anticipated Einstein. And then in 1904 he asked the most fundamental question of all: what is the shape of the space in which we live? It turned out to be possible to prove the Poincaré conjecture in unimaginable worlds, where objects have four or five or more dimensions, but not with three.
“The one case that is really of interest because it connects with physics, is the one case where the Poincaré conjecture hasn’t been solved,” said Keith Devlin, of Stanford University in California.
Mathematicians are optimistic that a potential solution proposed in 2002 by Grigori Perelman in papers posted to the internet may actually be correct, but there’s a snag:
Like Wiles, Perelman is claiming to have proved a much more complicated general problem and in the course of it may have solved a special one that has tantalised mathematicians for a century. But his papers made not a single reference to Poincaré or his conjecture. Even so, mathematicians the world over understood that he tackled the essential challenge. Once again the jury is still out: this time, however, his fellow mathematicians believe he may be onto something big.
The plus: the multidimensional topology of space in three dimensions will seem simple at last and a million dollar reward will be there for the asking. The minus: the solver does not claim to have found a solution, he doesn’t want the reward, and he certainly doesn’t want to talk to the media.
“There is good reason to think the kind of approach Perelman is taking is correct. However there are some problems. He is very reclusive, won’t talk to the press, has shown no indication of publishing this as a paper, which you would have to do if you wanted to get the prize from the Clay Institute, and has shown no interest in the prize whatsoever,” Dr Devlin said.
Which just goes to show that too much math will make you crazy. Or at least give you a major migraine.