Fermat's Last Theorem

The life of Sophie Germain

Timeline 1776Born on 1 April in Paris, the daughter of a wealthy silk merchant. 1789At 13, during the Revolution, she reads the story of Archimedes’ death and decides to study mathematics. 1794The École Polytechnique is founded and does not admit women. Sophie obtains the lecture notes and sends Lagrange a paper under the name Monsieur Le Blanc. 1801Gauss publishes the Disquisitiones Arithmeticae. 1804She begins writing to Gauss, again as Monsieur Le Blanc. 1806French troops occupy Braunschweig. Sophie asks a general who is a friend of the family to protect Gauss, who thus discovers who she is. 1809The Academy of Sciences announces a prize for the mathematical theory of vibrating plates. 1816On her third attempt, she wins the prize. She is the first woman to receive a prize from the Academy. 1823Legendre makes public her theorem on Fermat’s Last Theorem. 1829She learns she has breast cancer. 1831Dies on 27 June in Paris, aged 55. ◀ Previous Follow the journey Next ▶ 1776–1794 Rue Saint-DenisShe grows up in the house of her father, a rich silk merchant. At 13, during the Revolution, she decides to study mathematics. 1794 École PolytechniqueThe school does not admit women. She obtains the lecture notes and sends a paper to Lagrange as Monsieur Le Blanc. 1804–1806 BrunswickShe writes to Gauss about number theory. In 1806, with the city occupied, she asks a French general to protect Gauss, who thus learns who she is. 1816 Institut de FranceAt the third attempt she wins the Academy’s prize on vibrating plates, the first woman to receive a prize from the Academy. 1831 Père-Lachaise cemeteryShe dies on 27 June, aged 55. It is said that her death certificate recorded her as a rentière, a person of private means, and not as a mathematician. Enable JavaScript to see the map. The stages are listed below. Sophie Germain’s Paris, with a line for the letters sent to Gauss in Brunswick. Map: © OpenStreetMap contributors

October 14, 2026 Read

The letters of Monsieur Le Blanc: the arithmetic of remainders

When Sophie Germain wrote to Gauss in 1804, the two spoke a language that had just been born. Three years earlier, in the Disquisitiones Arithmeticae, Gauss had introduced a new sign, $\equiv$, and a new way of thinking about whole numbers: not by what they are, but by the remainder they leave on division. Everything Sophie later did on Fermat was written in that language. It is worth learning. Clock arithmetic If it is 9 o’clock now, what time will it be in 5 hours? Nobody answers 14 on a clock with hands: they answer 2. The clock keeps only the remainder on division by 12. Gauss wrote this as

October 14, 2026 Read

Sophie Germain's theorem and Fermat's Last Theorem

Around 1637 Pierre de Fermat noted in the margin of a book that the equation $$x^n+y^n=z^n$$has no solutions in positive whole numbers when $n$ is greater than 2, and that he had a marvellous proof which the margin was too narrow to contain. This is Fermat’s Last Theorem. Nobody ever found his proof, and the problem was only solved by Andrew Wiles in 1994. Until 1800 the situation was modest. Fermat had proved the case $n=4$, Euler the case $n=3$. It was enough to study prime exponents $n=p$, but each prime needed its own trick, and nobody could get past 3. Sophie Germain was the first person to prove something for infinitely many exponents at once.

October 14, 2026 Read

Sophie Germain primes and cryptography

A prime number $p$ is a Sophie Germain prime when $2p+1$ is also prime. 5 is one, because 11 is prime. 7 is not, because $15=3\cdot5$. The list begins: $$2,\ 3,\ 5,\ 11,\ 23,\ 29,\ 41,\ 53,\ 83,\ 89,\ 113,\ 131,\ldots$$The prime $2p+1$ that goes with each is called a safe prime: $5, 7, 11, 23, 47, 59, 83, \ldots$ Sophie Germain needed them for her theorem on Fermat. Two centuries later, they protect internet connections.

October 14, 2026 Read

The Sophie Germain identity: factoring what looks prime

Some numbers look prime and are not. 629, for example, is not even, does not end in 5 and is not a multiple of 3. But $629=5^4+4$, and that form gives it away: it factors by an identity that bears Sophie Germain’s name. The identity For any numbers $a$ and $b$, $$\begin{aligned}a^4+4b^4={}&\left(a^2+2b^2+2ab\right)\\&\times\left(a^2+2b^2-2ab\right).\end{aligned}$$The trick for finding it is completing the square, as in the quadratic formula. The sum $a^4+4b^4$ is almost the square of $a^2+2b^2$; only the middle term, $4a^2b^2$, is missing. Adding and subtracting that term,

October 14, 2026 Read

Chladni figures and the prize for vibrating plates

Sprinkle sand on a metal plate held at its centre and draw a violin bow across the edge. The plate begins to sing, and the sand, as if it had a will of its own, flees some places and gathers in others, drawing lines, crosses, stars and symmetrical curves. Change the note, and the drawing changes. These are Chladni figures. How a Chladni plate is excited: the plate is held at the centre and a violin bow is drawn across the edge. Engraving from an 1879 book.Unknown artist, 1879 · Wikimedia Commons · public domain Sound made visible The German Ernst Chladni published these figures in 1787, in a book called Discoveries on the Theory of Sound. The qualitative explanation is simple. When the plate vibrates, almost all its points move up and down and throw the sand away. But some lines stay still: these are the nodal lines. The sand slides there and stays. The drawing is a map of the places where the plate does not move.

October 14, 2026 Read

Mean curvature: how much a surface bends

To write the equation of vibrating plates, Sophie Germain had to answer a question that sounds simple: how much is a surface bent at a point? Her answer, mean curvature, is still used today to study soap bubbles, cell membranes and aircraft skins. The curvature of a curve Let us start with a line. A straight line does not bend, and we say its curvature is zero. A circle bends equally at every point, and the smaller it is, the more it bends. That is why the curvature of a circle of radius $R$ is defined as

October 14, 2026 Read

Sophie Germain's legacy

Algebra is nothing but written geometry; geometry is nothing but drawn algebra. — Sophie Germain, General Considerations on the State of the Sciences and Letters (1833) Sophie Germain worked almost always alone, without a degree, a post or students. She had no access to the lectures, journals or discussions through which the mathematicians of her time learned from one another, and this shows in her work: highly original ideas, sometimes with gaps a teacher would have pointed out in time. Even so, she left her name in two fields that barely touch, number theory and the theory of elasticity.

October 14, 2026 Read

Image credits for Sophie Germain

Historical images and photos All come from Wikimedia Commons. We use the thumbnails provided by Commons; the portrait was only converted from PNG to JPEG. Portrait of Sophie Germain, drawing by Berthe Chégaray of a medal in her memory, published in 1896: source page · public domain. Figures from the Research on the Theory of Elastic Surfaces, Sophie Germain, 1821, copy in the ETH-Bibliothek Zürich: source page · public domain. Chladni plate excited by a bow, engraving from 1879: source page · public domain. Grave in the Père-Lachaise cemetery, photo by Pierre-Yves Beaudouin, 2015: source page · CC BY-SA 4.0. Blog figures The cover, the nodal lines of a square plate and the drawing of the curvatures of the sphere, cylinder and saddle are original illustrations. The nodal lines were computed with the simplified model described in the text and do not reproduce photos of real plates.

October 14, 2026 Read