Closing in on Fermat's last theorem.

Closing in on Fermat's last theorem 

One of the most famous unsolved problems in mathematicsstarted out as a note scrawled in the margin of a book. Now known as Fermat's last theorem, the problem was first proposed by 17th-century French mathematician Pierre de Fermat. Although he had a wonderful proof for the theorem, Fermat wrote in the book's margin, he didn't have enough room to write it out. After Fermat's death, scholars could find no trace of the proof in any of his papers, and ever since, mathematicians have struggled in vain to solve the problem. Some recent mathematical discoveries, however, have tied Fermat's theorem more closely to modern mathematics, suggesting a possible avenue to a proof.

Fermat's conjecture (as it should properly be called until aproof is found) is related to a statement by the Greek mathematician Diophantus, who observed that there are whole numbers, x, y and z, that satisfy the equation x2 y2 = z2. For example, 3(2) 4(2) = 5(2). In fact, this equation has an infinite number of whole-number solutions. Fermat proposed that there are no solutions to the equation x(n) y(n) = z(n), when n is greater than 2. In other words, when n=3, no set of whole numbers satisfies the equation x3 y3 = z3, and so on.

A century after Fermat's death, Leonhard Euler proved thatFermat's conjecture is true for n=3. Later mathematicians found proof for other special cases, and a computer search performed a decade ago showed that Fermat's last theorem was true for all exponents less than 125,000. In 1983, Gerd Faltings, now at Princeton (N.J.) University, showed--as one consequence of his proof of the Mordell conjecture (SN: 7/23/83, p.58)--that if there are any solutions to Fermat's equations, then there are only a finite number of solutions for each value of n. But until recently, a proof for the general case has remained elusive.

The current effort to tame Fermat's conjecture started withthe work of Gerhard Frey of the University of the Saarlands in Saarbrucken, West Germany. Frey, who happened to be looking at equations for elliptic curves, written generally in the form y2 = x3 ax2 bx c, where a, b and c are constants, found a way to express Fermat's last theorem as a conjecture about elliptic curves. That put Fermat's problem into an area of mathematics where mathematicians have developed a wide range of tools and techniques for solving problems.

Frey wrote down the elliptic curve that would result ifFermat's conjecture were to be false. That elliptic curve turns out to have peculiar properties, and mathematicians examining the curve's equation had the feeling that the curve could not exist. If the curve's impossibility could be proven, then Fermat's conjecture would have to be true. Frey started on this task, but he failed to fill all the gaps in his attempted proof that the curve couldn't exist.

Then Jean-Pierre Serre of the College of France in Parissuggested that one of his own conjectures, if proven, would help patch up Frey's effort. Earlier this year, Kenneth A. Ribet of the University of California at Berkeley worked out the necessary proof of Serre's conjecture for a large class of situations. Ribet ended up showing that Fermat's theorem is true if certain elliptic curves arise from "cusp' forms.

Now Fermat's last theorem is tied to a central question innumber theory, known as the structural conjecture. That conjecture states that all elliptic curves arise from cusp forms. Most number theorists, for a variety of convincing reasons, believe the structural conjecture to be true, although it has not been proven. Even without a proof, this conjecture plays a major role in number theory. In fact, a proof that the conjecture is false would come as a shock to the mathematics community. Fermat's last theorem finally seems to rest on reasonably firm ground.