7 Days to Go: Sophie Germain

Sophie Germain has long been my favourite Mathematician. I find her stubbornness in practising Mathematics very inspiring and I have also always had a soft spot for Number Theory, which was her main area of work. She also has a fantastic name…

Marie-Sophie Germain was born on ${1}^{\text{st}}$ April $1776$ in Paris. She first picked up a Mathematics book aged $13$ , when she discovered J. E. Montcula‘s L’Histoire des Mathématiques. Her father was a representative of the bourgeoisie and so she was forced to stay inside due to the fall of the Bastille. She had sought entertainment in the library and from then on was hooked on Maths, reading all she could from the library. She even taught herself Latin and Greek to allow her to read more! Unfortunately this was not an endeavour her parents approved of, due to her being a woman. In an attempt to stop her studying, they didn’t allow her warm clothes or a fire at night. In response, Sophie wrapped herself in quilts, lit candles, and kept doing Maths.

The École Polytechnique opened in $1794$ , when Sophie $18$ . Once again being woman proved to be a barrier for Sophie, and she was not allowed to attend. And once again, this did not deter her. The lecture material was made publicly available, and students were invited to send in written observations. Under the pseudonym Monsieur Antoine-Augsute Le Blanc, Germain wrote in to Joseph-Louis Lagrange, a prominent Mathematician at the time. She also used this name to correspond with Adrien-Marie Legendre, and it was his Essai sur la théorie des nombres, published in $1798$ that piqued her interest in Number Theory. It was her exchanges with Carl Friedrich Gauss that were most famous though.

Germain was first driven to write to Gauss on reading his Disquisitiones Arithemticae. Over the course of several letters they discussed Gauss’ work and her thoughts on Fermat’s Last Theorem. Gauss initially knew Sophie as Monsieur Le Blanc, but her identity was rumbled during the Napoleonic wars. The French had occupied the German city of Braunschweig, where Gauss resided and Germain feared for his life. She was familiar with the legend of how Archimedes was killed during the siege of Syracuse, after being too absorbed in his Mathematics to meet with General Marcellus. In order to make sure Gauss didn’t suffer a similar fate, Sophie contacted General Pernety, a family friend of hers. He sent a chief of a battalion to meet with Gauss, but Gauss was very confused by the mention of Sophie’s name. Three months later she came clean and confessed her true name to him.

Germain thrice entered an contest hosted 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”. Despite being the only entrant, she was declined the prize on the first two occasions due to it being deemed that “the fundamental base of the theory [of elastic surfaces] was not established”. For her third attempt, Germain sought the advice of Poisson, who went on to publish his own work without acknowledgement of her contributions. Sophie still submitted her own paper, Recherches sur la théorie des surfaces élastiques, to the academy for consideration. Unfortunately Germain’s method relied on an incorrect equation from Euler so it did not predict experimental results with much accuracy, however she successfully derived the correct differential equation. On ${8}^{\text{th}}$ January $1816$ Germain became the first woman to win a prize from the Paris Academy of Sciences. Unbelievably, Despite she was still not allowed to attend sessions at the academy until later when Joseph Fourier, a good friend of hers, obtained tickets for her.

Having had a break from Number Theory to focus on the prize, Germain once again returned to the field and set about trying to prove Fermat’s Last Theorem. She outlined a general strategy for a proof, and even got as far as proving the theorem for all odd primes $p < 100$ . This work was later extended to primes $p < 1700$ by other Mathematicians.

Sophie continued to practise Mathematics until her death in $1831$ . Her name is immortalised in Number Theory as the eponym of Sophie Germain Primes: any prime $p$ such that $2p + 1$ is also prime.