The idle ramblings of a Jack of some trades, Master of none

Jan 17, 2008

It's Germane, Again

Recently, I wrote about Sophie Germain, acknowledged to the first modern female mathematician of the first rank. Until recently, her fame rested mainly on her work on Fermat's Last Theorem (FLT): a result1 by her, acknowledged by Legendre, was used to prove the veracity (by a brute force attack) of the FLT for the primes smaller than 100. It was generally assumed that she hadn't done much work in number theory other than this as very little of her published work was available for primary research.

All this changed some time ago when an entire archive of her correspondence and notes was discovered gathering dust at the Bibliothèque Nationale in Paris, and the Biblioteca Moreniana in Florence. Work by David Pengelley and others has revealed ambition, excellence and original technique in Germain, none of which was hinted at in any of the previously available literature.

In contrast to Legendre's approach towards the FLT, she had a novel method of attack - quite breath-taking in its scope at the time, although now known to be flawed. Remarkably, (as Laubenbacher and Pengelley state in their paper2), she makes no mention of the theorem that made her famous. Indeed, it appears from a letter she wrote to Gauss in 1819, her intention was to conquer FLT in one shot, rather than in the enumerative fashion that had been the vogue till then. As Laubenbacher and Pengelley say:
It requires that, for a given prime exponent p, one establish infinitely many auxiliary primes each satisfying a non-consecutivity condition on its p-th power residues (note that this condition is the very same as one of the two hypotheses required in Sophie Germain's Theorem for proving Case 1, but there one only requires a single auxiliary prime, not infinitely many). She writes that she has worked long and hard at this plan by developing a method for verifying the condition, made great progress, but has not been able to bring it fully to fruition (even for a single p) by verifying the condition for infinitely many auxiliary primes. She also writes that she has proven that any solutions to a Fermat equation would have to frighten the imagination with their size. And she explains in broad outline all her work on the problem.
To be fair, she did later realise the lacunae in her work, which were also pointed out by her friend, the mathematician Guglielmo Libri3 (of whom much, much more needs to be said). On the other hand, her programme shows much sophistication and a deep understanding of the subject. The tragedy, once again, was that despite her correspondence with some of the finest mathematicians of the time, she worked essentially in isolation. The likes of Legendre and Gauss who mentored her when she started tended to distance themselves as they moved onto other fields of research. Her impressive work does not seem to have been disseminated widely, and was unknown even to her friends. How much frustration she would have felt! What she would have achieved with the education and resources available to men of her generation! And yet she remained sanguine:
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.4


1. Sophie Germain's Theorem. For an odd prime exponent p, if there exists an auxiliary prime q such that there are no two nonzero consecutive p-th powers modulo q, nor is p itself a p-th power modulo q, then in any solution to the Fermat equation zp = xp + yp, one of x, y, or z must be divisible by p2.

2. “R. Laubenbacher, D. Pengelley, "Voici ce que j'ai trouvé:”Sophie Germain's grand plan to prove Fermat's Last Theorem". July 4, 2007.

3. A. Del Centina, A. Fiocca, Giunia Adini,and Maria Luisa Tanganelli, L'archivio di Guglielmo Libri dalla sua dispersione ai fondi della Biblioteca Moreniana = The archive of Guglielmo Libri from its dispersal to the collections at the Biblioteca Moreniana, L. S. Olschki, Firenze, 2004.

4. Sophie Germain, Letter to Gauss, 1819.


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