Beautiful Minds

After noticing on the calendar that it is March–Women’s History Month and seeing an old factorization method in my notes, I decided it was time to write about one of the most influential women in the world of mathematics. Despite the objectively greater prominence of men in terms of mathematical contributions, I want to highlight the genius of one particular woman back in the patriarchal days of society: Marie Sophie Germain, the inventor of modern number theory. 

Sophie Germain is one of the most accomplished yet underappreciated mathematicians of her time and is often considered the inventor of modern number theory. Indeed, her name is relatively obscure in the annals of mathematical history, and most people who may know of certain formulas she contributed to may not even know of her existence. It may come as a surprise that such a great contributor to a significant field of math like number theory is so unknown, but in the realm of mathematics, she is one of the most celebrated and decorated figures of all time. She was denied many fair learning opportunities because of her sex, such as being denied admission to the Polytechnic School of France, but persevered through adversities through her undying passion for mathematics. 

One of Germain’s greatest achievements was her contribution to the elasticity theory, which is a branch of physics and materials science that studies how a solid body deforms under applied stress and returns to its original shape once the stress is removed. She proposed a key hypothesis, that “elasticity was proportional to the sum of the inverses of the principal radii of curvature of a surface” (Tanzi). Although she never managed to convincingly prove or derive any mathematical equations inferred from the implications of her hypothesis, she was able to establish close relationships with influential mathematicians, with the likes of Lagrange and Gauss reaching out to her after her newfound progress. She maintained correspondence with these great mathematicians and scientists for the majority of her career. 

Germain also took significant steps to advance the proof of a specific case of Fermat’s Last Theorem, which is one of the most well-known theorems in number theory stating that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. Germain focused on substituting the exponent n for a prime number p, and proving that there were no non-zero integer solutions where p does not divide x, y, or z.  Fermat’s last theorem over the centuries has pushed many mathematicians to come up with more advanced techniques and substitutions, and Germain’s foundational proof was the foundation for all these newer ideas, and eventually, the complete proof. This has had a variety of meaningful impacts on modern-day society, especially regarding cryptography. 

Despite all these significant accolades, there is a bigger reason that she has had an impact on me and my interest in mathematics: her factoring identity of a^4+4b^4. Factoring higher-order polynomials requires a great deal of insight, casework, and forethought to see if all distributed multiplications and terms match up evenly. There are many factoring and expanding tricks that aspiring mathematicians learn, including FOIL (First Outer Inner Last) and Simon’s favorite factoring trick. Nevertheless, I would never have thought that a polynomial such as a^4+4b^4 could be factored in such a way, especially not by a mathematician from the 1700s who lacked the resources to study math efficiently on her own. Her factorization made me realize that in math there is almost always a solution and that what seems like a boring math problem without any words could still come together like an elegant puzzle. It was thanks to Germain’s discovery that I truly found the beauty of math and number theory and came to appreciate it. 

However, not everything went favorably for Germain in the latter part of her life. She developed breast cancer and died before she could be awarded or celebrated for many of her accomplishments. Gauss even proposed that she receive an honorary degree, but her death came before it could be bestowed. Nearing her death, Sophie Germain said, “Real superiority is nothing more than the means of considering difficult problems from a point of view whence they become easy, where the spirit can embrace them and follow them without effort.” 

Germain was a true pioneer in the world of mathematics who overcame immense societal and gender-based obstacles to usher in a paradigm shift that has had a lasting impact until today.

Works Cited

Barrow-Green, June. “Sophie Germain.” Encyclopædia Britannica, Encyclopædia Britannica, 2019, http://www.britannica.com/science/number-theory. Britannica.

Editors, Wikipedia. “Safe and Sophie Germain Primes.” Wikipedia, 18 Feb. 2024, en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes.

O’Connor, J. J., and E. F. Robertson. “Sophie Germain – Biography.” Maths History, 2020, mathshistory.st-andrews.ac.uk/Biographies/Germain/.

Tanzi, Cristina P. “Sophie Germain’s Early Contribution to the Elasticity Theory.” MRS Bulletin, vol. 24, no. 11, 1999, pp. 70–71, https://doi.org/10.1557/s0883769400053549.