Math Trek

An Attack on Fermat

Sophie Germain was the first to propose a realistic plan to prove Fermat's Last Theorem

3:41pm, February 21, 2008
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This is part one of a two-part series. Part II: "A Mathematical Tragedy" is available at About Time.

Around 1630, Pierre de Fermat scribbled his famous note in the margin of a book stating what is now known as "Fermat's Last Theorem." "I have discovered a truly remarkable proof which this margin is too small to contain," he added. His proof has never been found and was almost certainly wrong, but Fermat's conjecture bedeviled mathematicians for centuries to come.

Mathematicians soon realized that the problem was far harder than it first appeared. Number theorists labored endlessly to nibble off small parts of it, but in the early 1800s, one mathematician finally developed a bold strategy that had the potential to solve the whole problem at once. But the entire approach was very nearly lost to history, because until recently, all the notes and manuscripts were moldering unread in a French library.

The mathematician who developed the approach was respected by luminaries like Carl Friedrich Gauss, Adrien-Marie Legendre, and Joseph-Louis Lagrange, but was marginal in the mathematical community, with no formal training or university position. That's because the mathematician was a woman—indeed, the first woman to do significant research in mathematics.

Sophie Germain has been known for her work in the theory of elasticity and the curvature of surfaces, but until now, her only known work in number theory was a single result that Legendre attributed to her in a footnote.

"What he credited to her in this footnote is in some sense really a misrepresentation of what she did," says Reinhard Laubenbacher of Virginia Polytechnic and State University in Blacksburg. He and David Pengelley of New Mexico State University in Las Cruces searched through her notes in the Bibliothèque Nationale in Paris. Of the over 2,000 pages in the archive, hundreds and hundreds concerned number theory.

Some pages contained mere doodles that degenerated into chicken scratches, but many were filled with remarkable results. Included was a 20-page manuscript Germain had written so meticulously that not a single word was scratched out. "I personally believe," Pengelley says, "that she intended to submit it to the French academy for the prize for Fermat's Last Theorem."

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. (For n = 2, there are lots of solutions, for example, 32 + 42 = 52.) No complete solution to the problem was found until 1994, when Andrew Wiles of Princeton University cracked it using very sophisticated modern techniques from algebraic geometry.

During Germain's time, the main approach to the problem was to tackle it for particular exponents n, and it was known that it would suffice to prove the theorem for prime exponents. And Germain herself used the proof she has been known for, called Sophie Germain's Theorem, to show that the theorem is true for any prime n less than 100, if none of x, y, or z is divisible by n.

This result alone was remarkable, given the challenges Germain faced. As a woman, Germain couldn't enroll in the universities in France. So she took on the identity of a male student who had recently left the school, Antoine-August LeBlanc, reading lecture notes and sending in her homework assignments.

But somehow, her instructor, the great mathematician Lagrange, discovered her secret. According to a commentator at the time, Lagrange "went to her to express his astonishment in the most flattering of terms," and the commentator goes on to say that "the appearance of this young 'geomètre' made quite a stir." Nevertheless, the barriers against Germain's inclusion in the mathematical community didn't come tumbling down. She still couldn't enroll in the university, and her education was haphazard. And even after she had produced many impressive results, she had difficulty getting access to the French Academy of Sciences.

Indeed, Pengelley and Laubenbacher believe that she must have worked in far greater isolation than previously thought, judging from her notes. Legendre had been thought to have been a mentor for Germain. They both lived in Paris and both worked on Fermat's Last Theorem. But her notes show that they independently proved many of the same theorems and seemed to know little of one another's methods.

"Legendre's techniques were much more ad hoc," Pengelley says. "Germain would develop a theoretical approach or algorithm, not a computational one. She focused on methods of general applicability. She was more the theoretical mathematician." Pengelley and Laubenbacher presented their findings at the Joint Mathematics Meetings in San Diego in January.

Furthermore, Pengelley and Laubenbacher were astonished to find that Germain had something no other mathematician had at that time: a plausible, realistic plan for cracking Fermat's Last Theorem in its entirety, not just one number at a time, and not just if none of x, y, or z is divisible by n. This newly discovered grand plan used a completely different approach than what mathematicians have known from Sophie Germain's Theorem all this time.

"She was out to prove it all in one fell swoop," Laubenbacher says. "She was going to use this new math Gauss had developed. She read Gauss's book and said, 'That's the ticket to proving Fermat!' That was very bold. Nobody thought like that. Her role in 19th-century number theory was revolutionary."

This is part one of a two-part series. Part II: "A Mathematical Tragedy" is available at About Time.

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