Dec. 2, 2004 It's not as famous as Fermat's Last Theorem. In fact, the math problem, which has not had a correct solution since it was proposed in the 1960s, doesn't even have a name. But a new, elegant solution for the unnamed 40-year old problem has intrigued scientists enough to be published in a two-part paper in one of the world's top math journals.
Dino Lorenzini of the University of Georgia and two French colleagues, Michel Raynaud and Qing Liu, recently published the mathematical foundations for the new proof in the journal Inventiones Mathematicae and will soon published the proof itself.
"What makes this problem possibly unusual," said Lorenzini, "is that two mathematical mistakes in the literature had to be corrected before the problem could be correctly solved."
Intractable problems in mathematics have long had an allure for the general public. Fermat's Last Theorem, a significant hypothesis in number theory, was first stated by Pierre de Fermat, a 17th-century lawyer and amateur mathematician. The proposition was discovered by his son Samuel while collecting and organizing the elder Fermat's papers and letters posthumously. It took more than 350 years for mathematicians to solve the riddle, and that solution made front-page news worldwide.
The problem solved by Lorenzini and his colleagues is much more difficult to explain. In its simplest form, it is about understanding certain arithmetically interesting points (x,y) on a curve.
To study such a problem, mathematicians have ascribed to any curve C two special groups, called A and B, and first defined in the 1960s. Mathematical groups come in all sorts of shapes, sizes and patterns. Some are infinite, some are finite, some are big and some are small. They lie at the heart of attempts to classify what is going on in pure mathematics and in applications to chemistry, physics, biology and just about any subject using modern mathematics.
In modern number theory, one understands the solutions, in whole numbers or fractions, to equations by understanding, in part, the special groups that underlie them.
The conjecture is that the groups A and B always contain a finite number of elements. Despite the fact that top mathematicians believe this statement, until now, no one has been able to prove it, and this statement remains a major open problem in the field.
Lorenzini and his colleagues assumed that the number of elements in the group B is finite, and they were then able to demonstrate under this hypothesis that the number of elements in Group B can only be a perfect square. ("Square" here refers not to a geometric figure but to the product of whole numbers multiplied by themselves, as 1, 2, 3, 4 'squared' give 1, 4, 9 and 16, for instance.)
"Since mathematicians assumed that the groups A and B are finite, they have tried to compute the number of elements that such a group could contain," explained Lorenzini. "They believed in the early sixties that the number of elements in group A could only be a perfect square and, at the same time, they believed that the number of elements in group B wasn't always a perfect square."
This view held for some 30 years, but in 1996, the edifice started to crumble. The Japanese mathematician Toshuke Urabe revisited group B and found a mistake in previous work. Three years later, Bjorn Poonen of the University of California at Berkeley and Michael Stoll, a German scientist, found an example in which the number of elements in group A is not a perfect square.
"This was quite a reversal of fortune for these groups after 30 years," said Lorenzini.
Earlier this year, Lorenzini, Liu and Raynaud produced and published a precise formula relating the number of elements in groups A and B. Using this formula and the work of Poonen and Stoll, they were then able to show that the number of elements in group B is always a perfect square.
That part of the proof will be published soon, also in Inventiones Mathematicae.
While Lorenzini cheerfully admits that the new solution doesn't have the cachet of Fermat's Last Theorem and "isn't a Fields Medal winner" – the equivalent of a Nobel Prize in mathematics –the proof nonetheless is drawing delighted interest from mathematicians.
"What makes it interesting is the twist in the story," said Lorenzini. "It is not often in mathematics that two major assumptions on the same subject are corrected only after 30 years. But the real issue is that the groups A and B are so important and so difficult to study that any advance in this area is of interest to mathematicians."
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