It’s a well known fact of humanity that the chances of a group of people electing to do something decent and necessary is inversely proportional to the number of people in that group. We enshrine and attempt to forgive that principle under the banner of Institutional Inertia, but the fact remains that very decent individuals have, when given the power to act in concert, caused a great amount of pain in the history of science, with few examples as consistently pathetic as the ignored pleas of mathematician Sophie Germain (1776-1831) to have somebody, anybody, from Parisian academic circles acknowledge her theories publicly, or even provide feedback privately.
Germain’s life spanned France’s most turbulent political and intellectual years. Regicide, revolution, Empire, Restoration, the Hundred Days, another Restoration, and another revolution, all grinding past each other in dizzying succession while there gathered in Paris an All-Star roster of mathematical geniuses. Lagrange, Laplace, Fourier, Legendre, Monge, Cauchy, Abel, Galois, Dirichlet, Poisson – you can’t read through ten pages of a calculus or number theory text without running into something named after one of them, and they were all in the same city, Germain’s Paris.
We know precious little of her childhood except that, when revolution came in 1789, the young girl buried herself in the books of her father’s study, teaching herself mathematics and science while the old order crumbled just outside her window, learning enough Latin somewhere along the way to allow her to devour and understand central mathematical texts like Gauss’s landmark Disquisitiones arithemticae.
There was a limit to how far books and books alone could take her, however. To learn more, she needed access to the minds of the people who were pushing out the boundaries of mathematics at such a reckless clip. As a woman, she couldn’t attend their lectures, but she could write to them.
Worrying that they wouldn’t take a woman correspondent seriously, Germain pretended to be a Monsieur Leblanc and wrote to both Lagrange and Gauss, asking questions and forwarding mathematical proposals that favorably impressed them both. Her mind stood out even amongst her generation of mathematical geniuses, and when they eventually found out her secret, that she was a self-taught woman, their admiration only increased.
Her first area of interest was Number Theory, and in particular the problem we now know as Fermat’s Last Theorem, first proposed in the 17th century and only solved in the late 20th. This theorem says that you cannot find natural numbers x, y, and z such that xn + yn = zn for n > 2. For n=2 there are all sorts of natural numbers that work. You learn them in high school as the Pythagorean triplets: (3,4,5), (5,12,13), and so forth. But move just one power higher, Fermat theorized, and you won’t be able to find a single set of triplets no matter how long or how hard you search.
By Sophie Germain’s time, the theorem had been proven for n=3 and n=4, but a full solution for any value of n was proving elusive to the greatest mathematical minds of the age. Germain, of course, didn’t solve it either, but her approach to the problem, and the ideas that she developed in seeing it through, represent important advances in not only the solution of Fermat’s Theorem, but in the history of number theory. As such, her strategy deserves a bit of a closer look. So, nerd hats on, everybody!
Having solved the n=4 case, all that was really left was to show that the Theorem holds for any prime value of n. Why? Well, any natural number n greater than 2 must either be divisible by 4 or have an odd prime as a factor (think on that a while – it’s a neat little fact). For the former case, that means that we can rewrite xn + yn = zn as x4k + y4k = z4k and therefore as (xk)4 + (yk)4 = (zk)4, which is a form of the n=4 case which had already been proven to have no natural number solutions. For the latter case, if n has an odd prime factor, we’ll call it p, such that n = kp, we can rewrite the equation as xkp + ykp = zkp, and therefore as (xk)p + (yk)p = (zk)p, meaning that, really, the question we’re asking is whether natural number solutions exist when the power is prime.
Enter Germain. She had an exquisitely beautiful realization which picked up on Gauss’s use of modulo notation in the Disquisitiones. He had introduced a new type of congruence based on remainders whereby a and b are said to be “congruent modulo n” if they both have the same remainder when divided by n. 42 and 7 are both congruent modulo 5, for example, because both have a remainder of 2 when divided by 5. Germain took that idea and used it to construct a set of “auxiliary primes” out of each prime power p to be investigated.
To construct the auxiliary primes for a particular p, she looked at all the remainders when you divide xp by 2Np+1, where N is any natural number. If that set of remainders contained no consecutive numbers, she realized, then 2Np + 1 was an auxiliary prime for p and, most importantly, must divide either x, y, or z. To take the simplest non-proven case, that of p=5 and N=1, we get 2Np+1 = 11. When we repeatedly divide 15 through 105 by 11, the remainders we get are 1 and 10. Those aren’t consecutive, so 11 is an auxiliary prime of p=5, and therefore, if there is a triplet (x,y,z) that solved x5 + y5 = z5, one of those numbers must be divisible by 11.
Germain’s plan was to next prove that, for any prime p, there are an infinite number of auxiliary primes, and therefore that x, y, and z must have an infinite number of factors, which is clearly impossible. Thus no (x,y,z) triplet would exist for any prime power p, and since the primes are all that was needed to prove the problem generally, Fermat’s Last Theorem would have been put to rest. Unfortunately, proving that there are an infinite number of auxiliary primes was not something Germain, or anybody in her era, was equipped to do. The most she could do was to find enough auxiliary primes to show that, if a triplet existed, it must be insanely large, which seemed unlikely, but unlikely doesn’t mean impossible.
She didn’t prove Fermat’s Last Theorem, but she developed new conditions for its solution which broke the theorem into two sub-cases that later mathematicians could more efficiently whittle away at. In addition, she was the first mathematician to credibly forward a method that would attack the whole theorem at once, rather than just picking off single values of n here and there. In 1819, she sent her work to Gauss, hoping that the great reigning genius of number theory would advise her on how to proceed, but though Gauss had been friendly and encouraging when she had first written to him back in 1805, he did not, as far as we know, ever respond to Germain’s questions. Lacking outside help to push the problem through, she did not publish her results herself, and indeed it was only through word of mouth, and a footnote in one of Legendre’s books, that her significant contributions were known in France at all.
The refusal to publish makes sense when you look at it from the perspective of her other great contribution to mathematics, her differential equation for the elastic deformation of a disc. In 1808 a scientist and showman named Ernst Chladni came to Paris to show off his newest apparatus, a set of circular discs coated in a fine layer of dust that, when rubbed on the edge with a violin bow, formed distinctive patterns. The phenomenon fascinated everybody from society matrons to the Emperor Napoleon himself. In 1809 the Institut de France offered a prize to anybody who could mathematically explain the sand patterns that emerged from the vibration of the disc. Germain, who had studied the literature on vibrating bodies, decided to give it a shot and, indeed, hers was the only entry the Institut received, one which demonstrated a novel approach that made up for the errors in calculation that inevitably accompanied a problem so complicated.
The errors, however, were enough for the committee to not award her the prize. They announced a new competition with the same theme. She wrote up a new and improved version of her theory and, again, hers was the only submission. This time she got an honorable mention, but no prize. The committee then re-re-issued the challenge, and yet again Germain sent in her theory with further refinements and this time the prize was finally hers. In 1816 she became the first woman to receive a national prize for mathematics, but the victory was blunted by an unusually subdued and back-handed prize announcement which said, in effect, “Yeah, it’s an answer. It’s okay, we guess. I mean, we’ll give you the prize, if we have to.”
Germain understandably wanted to know just precisely what it was that the committee objected to in her work. She sent in a request for clarification which the members individually promised quick action on and, as a group, entirely ignored. Nobody would tell her what she had done wrong to receive such lukewarm praise surrounding her prize. About every two years thereafter, she rewrote her results and sent them to the Institut to be discussed and entered into the archives and, each time, with much individual assurance that it would be attended to right away, her papers were stuffed in a dark corner and forgotten. Nobody would tell her what was wrong or how she could improve, a situation bottomlessly irritating for somebody who wanted to get at the truth.
The habitual snubs of the academic community were oddly paired with her celebrity in the wider world. Winning the prize made her famous, somebody to know and to be seen to know. On a person to person level, her friendships were warm and encouraging. Fourier, Gauss, Lagrange, and Legendre were all friendly to her (though Poisson was a consummate ass who attempted to have her contributions written out of history). And yet, the institutions they largely ran repeatedly refused to officially evaluate and comment upon her work, letting it die unremarked or, at best, stuffed away in a footnote.
By the late 1820s, Germain had to curtail her mathematical activity in the face of a savage cancer that caused her too much pain to concentrate. She lived long enough to see the revolution of 1830, the second fall of the absolute school of monarchy, and also long enough to see the rise of a new generation of mathematicians, the wild and moody Galois, the tragic Abel, and the talented Dirichlet. Germain’s body died before her curiosity had run out, and that’s either quite tragic or extraordinarily beautiful, and most likely both.
FURTHER READING: Dora E. Musielak has written not only a fictional account of Sophie Germain, but a mathematically intense biography of her, Prime Mystery: The Life and Mathematics of Sophie Germain (2015). It has a significant number of typos, but doesn’t back down from the details of her work, which is refreshing. The level is a bit curious – it’s in exactly that zone where there’s too much detail for the scientifically literate amateur but not enough for the hard core mathematician meaning that, statistically, you’ll be frustrated one way or the other at some point. But this is a very necessary counterpoint to the Germain bashing that has gone on for so long, and that’s still in evidence even on her Wikipedia page, so by all means check it out.
Lead images via Wikimedia, public domain.