Today marks the 237th birthday of the French mathematician Sophie Germain (1776-1831). Germain was barely a teenager when the turmoil of the French revolution forced her to spend much of her time locked safely indoors. With not much else to do, Germain read her way through her father’s library and found herself deeply interested in the works on Archimedes, Sir Isaac Newton, and Leonhard Euler.

As you can imagine, Mathematics was not a proper subject for a young woman to learn, so she spent most of her life working by correspondence under the male pen name “Antoine-August Le Blanc.” Due to her interest in number theory inspired by Archimedes, she corresponded with many famous mathematicians specializing in number theory, such as Joseph Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss. She sent them her work and enticed so much interest from them that she eventually had to admit her true identity. Lagrange continued to be her mentor, while Gauss replied with the following remarkable quote:

“The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it. But when a person of that sex, that, because of our mores and our prejudices, has to encounter infinitely more obstacles and difficulties than men in familiarizing herself with these thorny research problems, nevertheless succeeds in surmounting these obstacles and penetrating their most obscure parts, she must without doubt have the noblest courage, quite extraordinary talents and superior genius.”

Germain later became interested in a contest by the Paris Academy of Sciences “to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence.” This was essentially a brand new field of mathematics and the amount of work and innovation involved scared away potential candidates. Germain was in fact the only participant in the competition. Despite her immense talent, her submission to the contest was riddled with errors due to her lack of formal education. The Academy extended the contest multiple times and Germain persisted. She continued to work on the problem and presented an improved solution at every iteration. Her persistence paid off, her third submission was accepted and she became the first woman to win a prize from the Academy in all of its long history beginning in 1666. However monumental the win, Germain still not allowed, as a woman, to attend sessions at the Academy until many years later due to a friendship with mathematician Joseph Fourier (Fourier Series, discovered greenhouse effects).

After the years spent working in the contest in elasticity theory, Germain returned her first love, number theory. More specifically, she focused on Fermat’s Last Theorem.

You may recall Fermat’s famous theorem which proposed, in 1637, that for any three integers *a*, *b*, and *c*, *a ^{n}* +

*b*=

^{n}*c*cannot be true for any integer value for

^{n}*n*greater than 2. Perhaps even more famous is Fermat’s note about his theorem: “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” The puzzle of discovering Fermat’s “marvelous proof” has been the world’s most difficult mathematics problem ever since.

Germain did the first major advancement on proving Fermat’s Last Theorem in 200 years since its inception. Prior to Germain, Fermat’s Last Theorem had only been proven by other mathematicians for *n* values 3, 5, and 7. Germain worked a subset of the original theorem, known as Sophie Germain’s Theorem:

“Let *p* be an odd prime. If there exists an auxiliary prime *P* = 2*Np* + 1 such that if *x ^{p}* +

*y*+

^{p}*z*= 0 (mod

^{p}*P*) then P divides

*x*

*y*

*z*, and

*p*is not a

*p*

^{th}power residue (mod

*P*). Then the first case of Fermat’s Last Theorem holds true for

*p*.”

Germain’s work proved the theorem for all primes smaller than 100. It wasn’t a full proof, but it was a big step forward.

While her work in elasticity was fundamental to the field, she remains most famous for her work on Fermat’s Last Theorem–in my book at least, but I have a soft spot for Fermat’s Last Theorem. Please join me today in celebrating Germain’s life and the advancement for women in mathematics. Happy birthday, Germain! May we all be as self-starting, persistent, and courageous as she was.

### Ariane Coffin

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