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Sophie Germain could not enrol at a university, never had formal training and was told by one astronomer to read a book for ladies with no equations in it. She still became one of the few people of her time to work seriously on both the physics of vibrating plates and the hardest problem in number theory. This post tells her story from reliable sources, explains her two big results, and works through the number theory with small primes you can check by hand.
How did Sophie Germain teach herself mathematics?
The MacTutor biography from the University of St Andrews records that she was born in Paris on 1 April 1776, the daughter of a prosperous merchant. She was 13 when the Revolution broke out. Kept at home, she read in her father's library and came across Montucla's history of mathematics, with its account of Archimedes so absorbed in a problem that he ignored the soldier who killed him. She decided she would become a mathematician.
Her parents were opposed. MacTutor tells how they took away her fire, her light and her clothes at night to stop her studying, and how she taught herself Latin and Greek so she could read Newton and Euler. They gave in after finding her asleep in the library one morning with the ink frozen in its well.
When the École Polytechnique opened in 1794, women could not enrol. Germain got hold of lecture notes, including Lagrange's analysis course, and sent him written observations under the name "M LeBlanc", borrowed from a former student. Lagrange was impressed enough to find out who had written them, and on learning she was a woman he became her sponsor and adviser.
How did she write to Gauss?
According to MacTutor, Germain studied Gauss's 1801 Disquisitiones Arithmeticae and, between 1804 and 1809, wrote him about a dozen letters, again as M LeBlanc. In 1806 French troops occupied Gauss's home town, Braunschweig. Remembering Archimedes' fate, she asked a French general who was a family friend to check on Gauss's safety. Gauss had never heard of her, and she had to reveal who she was. MacTutor reproduces his reply, which begins by describing his "admiration and astonishment" at the discovery.
What did she do in elasticity?
In 1808 the physicist Ernst Chladni showed Paris how sand on a vibrating plate settles into patterns, now called Chladni figures. As MacTutor tells it, the Institut de France set a prize for a mathematical theory of elastic surfaces that agreed with experiment. Most mathematicians did not try, because Lagrange had said the available methods were inadequate.
1811: Germain was the only entrant. She did not win, because she had not derived her hypothesis from physical principles.
Second round: again the only entry. She showed that an equation Lagrange had proposed reproduced Chladni's patterns in several cases, and received an honourable mention.
Contest reopened in 1815: her third entry won the prize, a gold medal, although gaps in rigour remained.
Poisson, her main rival on the subject and a judge, gave her little credit in public. A later paper she submitted in 1825 was ignored by the commission and only published in 1880. If the physics of stretching and bending is new to you, our guide to elasticity and Young's modulus covers the Class 11 basics.
What is a Sophie Germain prime?
Fermat's Last Theorem says xⁿ + yⁿ = zⁿ has no solution in positive whole numbers when n is greater than 2. MathWorld notes it was finally proved by Andrew Wiles in 1995, with a last step completed by Richard Taylor and Wiles. Long before that, Germain worked on the "first case", where the prime exponent p does not divide x, y or z. MathWorld's entry on the theorem states her result: she proved the first case for any odd prime p when 2p + 1 is also prime.
A prime p for which 2p + 1 is also prime is now called a Sophie Germain prime. The first few are 2, 3, 5, 11, 23, 29, 41 and 53. Nobody knows whether there are infinitely many.
In a letter to Gauss in 1819 she laid out a bigger plan. For each odd prime p, she wanted infinitely many "auxiliary" primes q such that the non-zero p-th powers, taken mod q, never include two consecutive numbers. Mathematicians Reinhard Laubenbacher and David Pengelley studied her manuscripts in detail in Historia Mathematica (2010). MacTutor notes the plan could not work in full: each p has only finitely many such primes. But it was the first attempt to prove the theorem for infinitely many exponents at once, and in 1985 a generalisation of her method proved the first case for infinitely many exponents.
Worked example: checking her condition for p = 3
MacTutor records that Germain showed 7 and 13 are the only auxiliary primes that work for p = 3. You can check both.
Cubes mod 7. For x = 1 to 6: 1, 8, 27, 64, 125, 216 leave remainders 1, 1, 6, 1, 6, 6. The non-zero cubes mod 7 are {1, 6}. No two are consecutive, so 7 passes.
Cubes mod 13. For x = 1 to 12 the remainders are 1, 8, 1, 12, 8, 8, 5, 5, 1, 12, 5, 12. The set is {1, 5, 8, 12}. Again no two are consecutive, so 13 passes.
Sophie Germain primes. 2 × 11 + 1 = 23 is prime, so 11 is one. 2 × 7 + 1 = 15 is not prime, so 7 is not.
Common mistakes: forgetting that 2 counts (2 × 2 + 1 = 5), working out x³ but not reducing it mod q, and thinking Germain's theorem proves the whole of Fermat's Last Theorem for those primes. It covers only the first case. Modular arithmetic like this appears in olympiad number theory, and primes themselves start in Class 10, as in our post on real numbers for Class 10.
What can a student take from Germain's way of working?
Use whatever access you can get. She had no classes, so she read lecture notes and wrote to the people who gave the lectures.
Treat a rejected attempt as feedback. Each prize entry improved on the last.
Work on more than one kind of problem. She moved between vibrating plates and prime numbers for most of her life.
When a topic in number theory or calculus has no one nearby to explain it, the Fermi Concept Library has every concept in mathematics with its own lesson, so you can fill one gap at a time.
List the cubes mod 19 and check whether any two non-zero ones are consecutive.
