Sophie Germain

How to use this course

Sophie Germain never attended a university, never held a post, and never had a formal teacher. She still produced the most important result on Fermat’s Last Theorem between Leonhard Euler and Ernst Kummer, and she won the Paris Academy’s prize for a mathematical theory of vibrating elastic plates. Both bodies of work were incomplete, and both were partly misjudged for a century and a half. Recent study of her manuscripts has shown that her number theory was far more ambitious than the single theorem that bears her name.

Lessons 04 to 07 follow her number theory, from the setting of Fermat’s problem to the modern theorems that grew out of her method. Lessons 08 to 10 follow her elasticity, from Chladni’s sand figures to the plate equation and the notion of mean curvature. Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or derivations, and exercises with folded solutions.

Timeline

Paris, 1776–1794

Marie-Sophie Germain was born in Paris on 1 April 1776 and baptised the same day. Her father, Ambroise-François Germain, was a wealthy silk merchant with liberal political views. In 1789 he was elected to the Estates-General, and later he became a director of the Bank of France. She had an older sister, Marie-Madeleine, and a younger one, Angélique-Ambroise. The household was comfortable and politically engaged, and it kept a good library.

She was thirteen when the Bastille fell. With Paris in turmoil she stayed at home and read. Her early biographers, starting with her friend Guglielmo Libri, tell how she found in Jean-Étienne Montucla’s Histoire des mathématiques the story of Archimedes, killed by a Roman soldier while absorbed in a geometric figure. A subject that could so possess a person seemed worth her life. The story is a family tradition rather than a documented fact, but her self-education is well attested.

Her parents thought mathematics unsuitable for a girl. The often repeated account says they took away her candles and her clothes at night so that she could not work, and that she was found in the morning asleep at her desk, the ink frozen in the well. Whatever the details, they relented. She taught herself Latin and Greek so that she could read Isaac Newton and Euler, and she worked through Étienne Bézout’s textbooks and Jacques Antoine Joseph Cousin’s calculus. Her father supported her financially for the rest of her life. She never married.

Monsieur Le Blanc, 1794–1804

In 1794 the revolutionary government founded the École Centrale des Travaux Publics, renamed the École Polytechnique in 1795. Lazare Carnot and Gaspard Monge were among its founders and Joseph-Louis Lagrange taught analysis there. Women could not enrol. Germain obtained the lecture notes of several courses and, at the end of a term, submitted written observations on Lagrange’s course under the name of a former student, Monsieur Le Blanc, who had left the school.

Lagrange noticed the quality of the work and asked to meet its author. On learning that Le Blanc was a young woman, he became her advocate and adviser and introduced her to other scientists. This was patronage, not training: she never received the systematic instruction that the École gave its students, and gaps in technique followed her through her career. Monge and Cousin also corresponded with her.

Number theory

In 1798 Adrien-Marie Legendre published his Essai sur la théorie des nombres. Germain studied it closely and began a correspondence with Legendre that grew into something close to collaboration. Legendre later included her results in his work and called one of her arguments very ingenious. In 1801 Carl Friedrich Gauss published the Disquisitiones Arithmeticae, which reorganised number theory around congruences. Germain mastered it, one of very few people in France who did so quickly, and on 21 November 1804 she wrote to Gauss, once more as Monsieur Le Blanc, enclosing results inspired by his book, including work on Fermat’s Last Theorem.

Gauss, Pernety and Chladni, 1804–1809

Gauss answered Le Blanc with respect and some praise, and he mentioned the correspondent favourably to colleagues. In October 1806 Napoleon defeated Prussia at Jena, and French troops occupied Brunswick, where Gauss lived. Germain remembered that Archimedes had died at the hands of a soldier. She wrote to General Joseph-Marie de Pernety, a friend of her family, asking him to make sure Gauss came to no harm. Pernety sent an officer to call on Gauss, who was puzzled to learn that his safety had been secured at the request of a Mademoiselle Germain whom he had never heard of.

She then explained, and Gauss’s reply of 30 April 1807 is the best-known letter in their exchange. He wrote that when a woman overcomes the obstacles placed in her way and penetrates the most obscure parts of number theory, she must have the noblest courage and “superior genius”. He also gave her number-theoretic claims honest scrutiny, at one point answering with a counterexample. In 1807 Gauss moved to Göttingen as director of the observatory, and his letters grew rarer. The correspondence ended in 1809. They never met.

A new problem

In 1808 the German physicist Ernst Chladni came to Paris and demonstrated the figures that sand makes on a vibrating plate: he clamped a glass or metal plate at its centre, drew a violin bow across its edge, and the sand gathered along the lines that did not move. Napoleon attended a demonstration, and in 1809 the First Class of the Institut proposed as its subject a mathematical theory of elastic surfaces compared with experiment. Lagrange is reported to have said that the problem needed a new kind of analysis. Germain decided to try.

Fermat’s problem and power residues

For the general reader. Pierre de Fermat claimed that \(x^n+y^n=z^n\) has no solution in positive whole numbers when \(n\ge3\). By 1800 the claim was proved only for \(n=3\) and \(n=4\). Germain’s idea was to study the equation not over the integers but modulo a well-chosen auxiliary prime, where the \(n\)th powers form a small, rigid set. This lesson sets up that language.

Reductions

If \(x^{mn}+y^{mn}=z^{mn}\), then \((x^m)^n+(y^m)^n=(z^m)^n\). Every \(n\ge3\) is divisible by 4 or by an odd prime, so the theorem follows from the cases \(n=4\) (Fermat, by infinite descent) and \(n=p\) an odd prime. Write the equation symmetrically as \(x^p+y^p+z^p=0\) (replace \(z\) by \(-z\), allowed since \(p\) is odd), and assume \(x,y,z\) pairwise coprime.

Power residues

Let \(\theta\) be a prime. The group \((\mathbb Z/\theta)^\times\) is cyclic of order \(\theta-1\), as Gauss proved in the Disquisitiones. A nonzero residue \(a\) is a \(p\)th power residue if \(a\equiv u^p\) for some \(u\).

Proof. The map \(u\mapsto u^p\) on a cyclic group of order \(\theta-1\) has kernel of size \(\gcd(p,\theta-1)\). If \(p\nmid\theta-1\) it is a bijection. If \(\theta-1=2Np\), its image has index \(p\), so it is the unique subgroup of order \(2N\). Since \(p\) is odd and \(2N\) is even, \(-1\) is always a \(p\)th power residue in this case.

Sophie Germain’s theorem

For the general reader. Germain found a test that, when an auxiliary prime with two simple properties exists, rules out every solution of Fermat’s equation in which the exponent does not divide any of the three numbers. Legendre published the result in 1823 with full credit to her, and for a century it was the main general tool on the problem.

Proof

Suppose \(x^p+y^p+z^p=0\) with \(x,y,z\) pairwise coprime and \(p\nmid xyz\). The hypotheses force \(\theta\equiv1\pmod p\) (Exercise 4.3; \(\theta=2\) fails hypothesis 2), so \(\theta>p\).

Step 1: factor. Write \(-x^p=y^p+z^p=(y+z)\,\varphi(y,z)\) with \(\varphi(y,z)=y^{p-1}-y^{p-2}z+\cdots+z^{p-1}\). Modulo \(y+z\) we have \(z\equiv-y\), so \(\varphi(y,z)\equiv p\,y^{p-1}\). Hence any common prime factor of \(y+z\) and \(\varphi\) divides \(p\,y^{p-1}\); it cannot divide \(y\) (else it divides \(z\)), so it is \(p\). But \(p\nmid x\), so \(\gcd(y+z,\varphi)=1\), and both factors are \(p\)th powers (signs can be absorbed since \(p\) is odd):

Step 2: \(\theta\) divides one of \(x,y,z\). If not, divide \(x^p+y^p\equiv-z^p\) by \(x^p\): \(1+(y/x)^p\equiv(-z/x)^p\pmod\theta\). Then \(r=(y/x)^p\) and \(r+1=(-z/x)^p\) are consecutive nonzero \(p\)th power residues, contradicting (N-C). This step uses only \(x^p+y^p+z^p\equiv0\), not coprimality. By symmetry we may assume \(\theta\mid x\).

Step 3: \(\theta\mid a\). Now \(2x=b^p+c^p-a^p\equiv0\), so \(b^p+c^p+(-a)^p\equiv0\pmod\theta\), and Step 2 again shows \(\theta\) divides one of \(a,b,c\). If \(\theta\mid b\), then \(z\equiv-x\equiv0\), contradicting \(\gcd(x,z)=1\); likewise \(\theta\mid c\) is impossible. So \(\theta\mid a\), that is \(y\equiv-z\pmod\theta\).

Step 4: contradiction. Then \(\alpha^p=\varphi(y,z)\equiv p\,y^{p-1}\pmod\theta\). Also \(x+y=c^p\) with \(\theta\mid x\) gives \(y\equiv c^p\). Therefore \(p\equiv\alpha^p\,c^{-p(p-1)}=\big(\alpha c^{1-p}\big)^p\), and \(\theta\nmid c\) because \(\theta\nmid y\). So \(p\) is a \(p\)th power residue modulo \(\theta\), contradicting hypothesis 2. ∎

Auxiliary primes and the grand plan

For the general reader. Germain did not aim only at Case I. Her manuscripts, studied in detail by Reinhard Laubenbacher and David Pengelley in 2010, show a plan to prove the whole theorem: find infinitely many auxiliary primes that must divide a solution, so that no solution can exist. The plan fails for a deep reason, and she found the failure herself for \(p=3\).

Germain primes

Proof. With \(N=1\) the nonzero \(p\)th power residues are the solutions of \(a^2\equiv1\), namely \(\pm1\). A consecutive pair would be \(\{-1,0\}\) or \(\{1,2\}\); the first contains 0 and the second needs \(2\equiv-1\), i.e. \(\theta=3\). Finally \(p\equiv-1/2\pmod\theta\), which equals \(\pm1\) only if \(\theta\mid 1\) or \(\theta\mid3\). ∎

Primes \(p\) with \(2p+1\) prime are now called Sophie Germain primes. Whether there are infinitely many is still open.

The plan

In Step 2 of Lesson 05 only condition (N-C) was used to show \(\theta\mid xyz\). Germain observed that if (N-C) held for infinitely many \(\theta\), the integers \(x,y,z\) would have infinitely many prime factors, which is absurd. This would prove the full theorem for \(p\), both cases at once. Her manuscripts develop methods to establish (N-C) for \(\theta=2Np+1\) for whole families of \(p\) at a time, with \(N=1,2,4,5,7,8,10\), and her letter to Gauss of 12 May 1819 describes the programme.

The modern explanation is geometric. Consecutive nonzero \(p\)th power residues are points on the Fermat curve \(u^p-v^p=1\) over \(\mathbb F_\theta\) with \(uv\ne0\). By the Hasse–Weil bound (proved by André Weil in 1948) this curve has \(\theta+O(p^2\sqrt\theta)\) points, so for \(\theta\) large compared with \(p^4\) such points must exist. Germain proved by hand that for \(p=3\) the only primes satisfying (N-C) are 7 and 13, and sent the argument to Legendre. A computer search shows that for \(p=5\) the list is \(11, 41, 71, 101\).

From Legendre to Wiles

For the general reader. Germain’s theorem was published inside a memoir by Legendre, and for a long time her name survived mainly through his footnote. Her method kept working long after her death: it was extended by hand to thousands of exponents, and in 1985 a refinement combined with modern analytic number theory proved Case I for infinitely many primes.

The nineteenth century

Legendre’s 1823 memoir on Fermat’s theorem stated Germain’s theorem, reproduced her table of auxiliary primes for every odd \(p<100\), and extended the verification of Case I to all \(p\le197\). In 1825 Peter Gustav Lejeune Dirichlet and Legendre proved the full theorem for \(p=5\). Gabriel Lamé settled \(p=7\) in 1839. Ernst Kummer’s ideal numbers (1847) proved the theorem for all regular primes, a completely different method. In 1908 Leonard Eugene Dickson pushed Germain-type criteria to prove Case I for all \(p<1700\).

Infinitely many exponents

The generalized criterion asks for primes \(\theta=2kp+1\) with conditions on \(k\); if \(k\) can be kept small relative to \(p\), the criterion applies. Showing that such \(\theta\) exist for infinitely many \(p\) is a question about primes in arithmetic progressions, a sieve problem.

Leonard Adleman and Roger Heath-Brown reduced the problem to an estimate on primes \(q\) for which \(q-1\) has a very large prime factor \(p\); Étienne Fouvry supplied the estimate through an improved Brun–Titchmarsh inequality. A decade later Andrew Wiles, with Richard Taylor, proved the whole theorem by the modularity of semistable elliptic curves (1995), by a route with no direct link to Germain’s.

Vibrating plates

For the general reader. A violin string is described by one equation that Jean le Rond d’Alembert wrote in 1747. A flat plate is harder because it resists bending in two directions at once. The prize Germain entered asked for its equation. Lagrange, a judge, wrote it down in 1811 after correcting her calculation, and it is still the equation engineers use.

Lagrange wrote the equation in the form \(\partial_t^2z+k^2\nabla^4z=0\), with the constant \(k^2\) left undetermined; its value in terms of material constants came later, from Claude-Louis Navier (1820) and Gustav Kirchhoff (1850). The fourth-order operator is the two-dimensional version of the beam equation \(EI\,w''''+\rho A\,\ddot w=0\) of Daniel Bernoulli and Euler.

Modes of a rectangle

For a plate \(0\le x\le a,\ 0\le y\le b\) with simply supported edges (\(w=0\) and \(\nabla^2w=0\) on the boundary), separation of variables gives

Frequencies grow like the square of the mode number, unlike a string or membrane, where they grow linearly. The nodal lines are where \(w_{mn}\) vanishes. Chladni’s plates were free at the edges and clamped at the centre, which changes the boundary conditions and makes the modes harder to compute, but the same mechanism produces his figures: when two modes share a frequency, the plate vibrates in a combination whose nodal set is curved.

Bending energy and boundary conditions

For the general reader. Germain’s central idea was physical: a bent plate resists in proportion to how much its curvature has changed, measured by the sum of the two principal curvatures. That idea gives the right equation inside the plate. What went wrong in her prize essays was the behaviour at the edges, and the reason was only clarified by Kirchhoff in 1850.

Germain’s hypothesis

For a nearly flat plate \(z=w(x,y)\), the principal curvatures \(\kappa_1,\kappa_2\) satisfy \(\kappa_1+\kappa_2\approx\nabla^2w\) and \(\kappa_1\kappa_2\approx w_{xx}w_{yy}-w_{xy}^2\). Germain postulated that the elastic restoring force depends on \(\kappa_1+\kappa_2\), by analogy with the beam, whose resistance depends on its single curvature. In energy terms her hypothesis corresponds to

Varying \(w\mapsto w+\varepsilon\eta\) and integrating by parts twice,

So the interior equation is \(D\nabla^4w=q\) for a load \(q\), Lagrange’s operator.

The second term is a null Lagrangian: \(w_{xx}w_{yy}-w_{xy}^2=\partial_x(w_xw_{yy})-\partial_y(w_xw_{xy})\) is a divergence, so it changes only boundary terms. This explains the history. Germain, Lagrange and later Navier all had the correct interior equation, because it does not see the Gaussian term. The edge conditions do, and those were wrong in Germain’s essays. Siméon Denis Poisson in 1829 imposed three conditions at a free edge, one too many for a fourth-order equation; Kirchhoff showed that two of them combine into the single “effective shear” condition above. Her energy \(U_G\) is the special case \(\nu=1\).

Mean curvature

For the general reader. A surface can curve differently in different directions: a saddle bends up one way and down the other. Germain needed a single number to measure bending at a point, and she chose the average over all directions. That number, the mean curvature, is now central in geometry, from soap films to general relativity. Her last memoir, published in 1831, was devoted to it.

At a point of a smooth surface with unit normal \(n\), each tangent direction \(e\) has a normal curvature \(\kappa(e)\), the curvature of the normal section in that direction. Leonhard Euler showed in a memoir published in 1767 that \(\kappa\) has a maximum \(\kappa_1\) and minimum \(\kappa_2\) in orthogonal directions.

Gauss’s Disquisitiones generales circa superficies curvas (1827) showed that \(K\) is intrinsic: it can be measured inside the surface. \(H\) is not intrinsic; a flat sheet and a cylinder have the same \(K=0\) but different \(H\). That is why \(H\) is the right quantity for bending a plate: bending a sheet into a cylinder costs energy even though no distances on the sheet change. For a graph \(z=w(x,y)\),

Surfaces with \(H=0\) are minimal surfaces, the shapes of soap films studied by Lagrange and Jean Baptiste Meusnier. In 2023 the engineer Douglas Holmes argued that \(H\) should be called the Germain curvature.

The prize and after, 1809–1829

The elasticity prize had a deadline of October 1811. Most mathematicians stayed away, and Poisson, who might have competed, was elected to the Institut in 1812 and became a judge. Germain’s memoir was the only entry. It started from the right hypothesis but contained errors in the calculus of variations, and the judges ruled that the true equations had not been established. Lagrange, working from her hypothesis, derived the correct equation himself. The prize was set again. Her 1813 entry, again the only one, showed that Lagrange’s equation reproduced several of Chladni’s figures but did not derive it convincingly from physical principles; she received an honourable mention.

In 1814 Poisson presented his own memoir on elastic surfaces, based on a molecular model, without acknowledging her. Her third entry won the prize on 8 January 1816, the first time a woman won a prize of the Paris Academy. The judges still had reservations about her derivation and boundary conditions. She did not attend the ceremony, perhaps sensing those reservations. In 1821 she published the work at her own expense as Recherches sur la théorie des surfaces élastiques, partly to answer Poisson. A further memoir sent to the Academy in 1825 was set aside by the commission and only published in 1880, after it was found among Gaspard de Prony’s papers. Her 1826 Remarques received a favourable report from Augustin-Louis Cauchy.

Recognition came slowly and partially. Joseph Fourier, permanent secretary of the Academy from 1822, obtained tickets that allowed her to attend its public sessions. In 1815 the Academy proposed a prize for Fermat’s Last Theorem, which drew her back to number theory and to Gauss, with no reply. Legendre’s 1823 memoir credited her theorem. She formed friendships with Fourier and, in her last years, with the young Italian mathematician Guglielmo Libri.

Illness, death, legacy

In 1829 Germain learned that she had breast cancer. She continued to work on number theory and on curvature, and wrote on philosophy. Her Mémoire sur la courbure des surfaces appeared in August Leopold Crelle’s journal, dated 1831. She died on 27 June 1831 at 13 rue de Savoie in Paris, aged 55, and was buried in the Père Lachaise cemetery. Her death certificate listed her not as a mathematician but as a rentière, a woman of independent means.

Her nephew Armand-Jacques Lherbette published her philosophical essay Considérations générales sur l’état des sciences et des lettres in 1833. It argues that the sciences and the humanities share the same aims and methods, and Auguste Comte praised it. In 1837, at the centenary of the University of Göttingen, Gauss said that she had deserved an honorary degree and regretted that she had not lived to receive one.

Legacy

• Number theory: Sophie Germain’s theorem, Sophie Germain primes (now used in cryptography, where \(2p+1\) is a “safe prime”), and a programme for the whole of Fermat’s Last Theorem that the 2010 study of her manuscripts restored to view.
• Elasticity: the bending hypothesis behind the plate equation, a foundation of structural mechanics. Her name is not among the 72 scientists engraved on the Eiffel Tower, an omission long criticised; Paris has announced plans to add the names of women scientists, hers among them.
• Geometry: the use of mean curvature as the measure of bending.
• Honours: the rue Sophie Germain and the Lycée Sophie Germain in Paris, a plaque at her death house, and since 2003 the Prix Sophie Germain of the Académie des sciences.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Sophie Germain, MacTutor History of Mathematics.
2. Sophie Germain, Wikipedia.
3. E. E. Kramer, Germain, Sophie, Dictionary of Scientific Biography.
4. L. L. Bucciarelli and N. Dworsky, Sophie Germain: An Essay in the History of the Theory of Elasticity, Reidel, 1980.
5. D. Musielak, Prime Mystery: The Life and Mathematics of Sophie Germain, AuthorHouse, 2015.
6. H. Stupuy (ed.), Œuvres philosophiques de Sophie Germain, Paris, 1879.

Mathematics

1. R. Laubenbacher and D. Pengelley, “Voici ce que j’ai trouvé”: Sophie Germain’s grand plan to prove Fermat’s Last Theorem, Historia Mathematica 37 (2010).
2. A.-M. Legendre, Recherches sur quelques objets d’analyse indéterminée et particulièrement sur le théorème de Fermat, Mém. Acad. Sci. Inst. France 6 (1823).
3. S. Germain, Recherches sur la théorie des surfaces élastiques, Paris, 1821.
4. S. Germain, Mémoire sur la courbure des surfaces, J. reine angew. Math. (1831).
5. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. reine angew. Math. 40 (1850).
6. L. M. Adleman and D. R. Heath-Brown, The first case of Fermat’s last theorem, Invent. Math. 79 (1985).
7. É. Fouvry, Théorème de Brun–Titchmarsh; application au théorème de Fermat, Invent. Math. 79 (1985).
8. D. P. Holmes, Germain curvature: the case for naming the mean curvature of a surface after Sophie Germain, arXiv:2303.13615.
9. P. Ribenboim, Fermat’s Last Theorem for Amateurs, Springer, 1999.
10. I. Todhunter and K. Pearson, A History of the Theory of Elasticity, vol. 1, Cambridge, 1886.