Sunday, March 9, 2008

An Attack on Fermat, Part 2

A Mathematical Tragedy

Sophie Germain had a bold program to prove Fermat's Last Theorem

This is part two of a two-part series. Part I: "An Attack on Fermat"

Nearly two centuries ago, Sophie Germain, the first woman known to have discovered significant mathematical theorems, developed a bold plan to prove Fermat's Last Theorem. But this entire plan was nearly lost to history, until David Pengelley of New Mexico State University in Las Cruces and Reinhard Laubenbacher of Virginia Tech in Blacksburg dug through her notes, long archived in a French library.

Fermat made his conjecture in 1630, but it took more than 350 years for mathematicians to finally come up with a proof of it. Andrew Wiles of Princeton University cracked the problem in 1995. In Germain's day, almost all mathematicians working on the problem tackled only small bits of it at a time. But Germain's approach, had it been successful, would have proven the entire conjecture at one go. Because her work was almost entirely unknown, mathematics ended up reproving some of her results 80 years later.

Before Pengelley and Laubenbacher's recent discoveries, mathematicians knew only of a small partial result of Germain's in number theory. But in her manuscripts, they found a simple, direct plan of attack on Fermat's entire theorem. She exploited techniques developed by Carl Friedrich Gauss and laid out her method in a letter to him in 1819, looking for feedback and, perhaps, collaboration.


Sophie Germain (1776–1831) was the first woman known to have discovered significant mathematical theorems.

She had initially written to the master mathematician in 1804, using her male pseudonym of Antoine-August LeBlanc. She shared with Gauss some proofs that grew from her reading of his great work Disquisitiones Arithmeticae. He had responded with enthusiasm, saying "it pleases me that arithmetic has acquired in you so able a friend." Their correspondence continued for 4 years.

Eventually, Gauss discovered her secret. In 1806, Napoleon's armies were marching into Prussia, and Germain became concerned that Gauss might be in danger. She asked a friend who was a commander in the French artillery to find Gauss and ensure his safety. Her friend followed her request—but revealed her identity in the process.

Gauss initially responded with delight, writing to Germain: "The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare.… But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius."

Yet, in his next letter, Gauss closed their correspondence, saying he had a new job in astronomy and would no longer have time for her mathematical investigations. She heard from him only once more, when his assistant wrote asking for her help in selecting a clock as a gift from Gauss to his wife.


Sophie Germain corresponded for four years with Carl Friedrich Gauss, one of the greatest mathematicians of all time.

Germain seems to have accepted Gauss's closure, though it isn't known whether she helped with the clock. But she didn't stop her work on number theory. And when she developed her method to solve Fermat's Last Theorem, she wrote again.

Fermat's Last Theorem states that there are no nonzero whole numbers x, y, and z such that xn + yn = zn for any n greater than 2. Germain's approach to proving it used Gauss's new technique of modular arithmetic, which divides different numbers by some fixed number and only considers the remainder. So, for example, 4 modulo 3 is 1, and 7 modulo 3 is also 1.

Germain's idea was that it is often easier to prove that Fermat's equation can't be true when taken modulo some "auxiliary prime" P. It was a promising approach, but one aspect of it was challenging. A step in her method was to divide both sides of the equation by x. But if it happened that P divided x evenly with no remainder, then x would be equal to 0 modulo P. And of course, in mathematics, it isn't legitimate to divide by zero. For related reasons, neither y nor z could be evenly divisible by P either.

So if she proved that Fermat's equation didn't hold modulo P, it only showed that the regular, non-modular version didn't hold for x, y, and z that aren't divisible by P. But that wasn't good enough—she needed to prove it with no restrictions.

So Germain devised a way around the problem. She noted that for any particular values of x, y, and z, only finitely many numbers could possibly divide any of them evenly. So she just had to prove that she could carry her method above through for infinitely many values of P. Then, for any values of x, y and z, she would know that for some P, she'd have proven Fermat's equation couldn't be true modulo that number. That would mean that Fermat's equation wouldn't hold for any particular choice of x, y and z, and Fermat's Last Theorem would be proven.

When Germain wrote to Gauss, she knew she had not yet completed the project. "I have never been able to arrive at the infinity, although I have pushed back the limits quite far by a method of trials too long to describe here," she wrote. "You can easily imagine, Monsieur, that I have been able to succeed at proving that this equation is not possible except with numbers whose size frightens the imagination.… But all that is still not enough; it takes the infinite and not merely the very large."

Gauss didn't respond to her letter.

Germain never managed to arrive at infinity, though she pushed her approach a long way. But the effects of her isolation show in her work. Scattered throughout are little mistakes. "We all make mistakes, and colleagues or referees catch them," Laubenbacher says. "She didn't get that."

Pengelley agrees. "I think what the mistakes show more than anything else is that she didn't have other people who read her work and gave her feedback. It's conceivable to me—unbelievably—that some of her most important manuscripts might not have been read by anybody." Pengelley and Laubenbacher presented their findings at the Joint Mathematics Meetings in San Diego in January.

Even without the mistakes, though, her program could not have succeeded. The problem was simply too hard, and nearly 200 years of mathematical development would be needed before it could be cracked. Eventually, she herself proved that her approach couldn't work.

Germain might have been able to accept the overwhelming difficulty of the problem. "I have never ceased thinking about the theory of numbers," she wrote in her letter to Gauss explaining her program. "I will give you a sense of my absorption with this area of research by admitting to you that even without any hope of success, I still prefer it to other work which might interest me while I think about it, and which is sure to yield results."

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