What the work claims
That a differential equation could model elastic surface vibration, and that Fermat’s Last Theorem’s first case holds under certain modular conditions for many odd primes.
How it was done
Germain developed a mathematical theory of vibrating elastic surfaces and submitted it as an essay to the Paris Academy of Sciences. She also devised a theorem for Fermat’s Last Theorem using modular arithmetic and congruence conditions.
What holds up
She derived the correct differential equation for vibrating elastic surfaces—a special case of the Kirchhoff–Love equation. She proved the first case of Fermat’s Last Theorem for all odd primes less than 100, and possibly up to 197. Her theorem remains named and cited.
What does not
Her elasticity theory did not accurately predict experimental results because she used Euler’s incorrect equation, leading to wrong boundary conditions. Her Fermat work did not prove the full theorem—only the first case for specific odd primes.
Why it matters beyond the lab
Her elasticity work founded a branch of mathematical physics still used in engineering. Her number-theoretic theorem shaped how mathematicians approached Fermat’s Last Theorem for over two centuries—before Wiles’ proof—by narrowing viable strategies.
Is it worth your time
Yes—if you care about how foundational mathematical work can be both correct in its core formalism and flawed in its physical application, or how a single theorem can structure centuries of proof attempts without resolving the original problem.