Sophie Germain and Fermat’s Last Theorem

Sophie Germain-1776-1831-grangerOne doesn’t often meet professional mathematicians so I am reposting this piece (which I originally wrote for the OSU history of science blog) in honor of Matt Klassen, the head of the mathematics department at DigiPen Institute of Technology. I met Matt, who also happens to be a fine guitarist, at David Russell’s masterful guitar concert in Seattle last weekend. Dr. Klassen has presented on “Non-Associative Loops on Fermat Curves of Odd Degree,” which reminded me of the French mathematician, Sophie Germain (1776-1831):

On May 20, 2014, the OSU Department of Mathematics sponsored a history lecture by Dr. David Pengelley, of New Mexico State University. Dr. Pengelley presented an animated lecture on some of the work of Sophie Germain. Dr. Pengelley’s interest in Germain was sparked by his use of primary historical sources in his teaching of mathematics. This led him to the National Library of France (Bibliothèque nationale de France) where he found a store of Germain’s original manuscripts which had not been studied in over two hundred years. Revisiting Germain’s work as a mathematician, Dr. Pengelley found that Germain had developed a sophisticated plan for proving Fermat’s Last Theorem, making significant contributions to number theory. Until recently, her work was known only via a footnote in another mathematician’s treatise (Legendre, Essai sur la Théorie des Nombres, 1823). In an age when women were usually not well-educated and when they were explicitly excluded from scientific academies, Germain’s substantial achievements were indeed remarkable.

Sophie Germain was only thirteen when the French Revolution broke out, forcing her to spend most of her time indoors. During that period, she turned to her father’s library. Fascinated by books on mathematics, she taught herself against her parent’s wishes (Pengelley relates that at one point they even took away her clothes and candles to prevent her from studying at night!). Germain’s father was a silk merchant so it was not through his mentorship that she developed her abilities but rather through her own effort and perseverance. At one point, Germain took on the identity of a student at the École Polytechnique who had died (Antoine-August LeBlanc). When the professor discovered that it was really a woman who was submitting such fine work under LeBlanc’s name, he was astonished. Germain eventually corresponded with Johann Carl Friedrich Gauss (1777-1855) in Göttingen, one of the most celebrated mathematicians of the time. Pengelley recounts that upon receiving a letter from Germain, Gauss praised the way she contributed to the “charms of this sublime science,” as giving him great joy.

Pengelley gave a cogent and fairly detailed explanation of the theorem by Pierre de Fermat (c.1601-1665) that Germain was hoping to prove. Basically, the theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer n greater than two. At the time that Germain was working on the problem, it was known that the theorem could be proven to hold for some numbers but much work remained before the theorem could be proven conclusively. Germain’s letters and manuscripts demonstrate that she had a good handle on the problem and that she had made considerable progress toward a solution. Pengelley found that she had made a mistake in one of her proofs but peering closer found scribbled in the margins, “voyez errata”—Germain’s own admission that she saw she had made an error!

Germain did win a prize from the French Academy of Sciences for her work on elasticity and she eventually was able to attend the Society’s meetings, but she was never made a member nor was any of her work published. Her manuscripts were taken by Guillaume Libri, described by Pengelley as a “mathematician, historian, bibliophile, thief, and friend of Sophie Germain.” Because Libri ended up with her manuscripts, they were preserved and eventually made available for Pengelley’s research. Finding a proof for Fermat’s theorem has been a problem that has long attracted the attention of mathematicians. In 1995, the mathematician Andrew Wiles, with help from Richard Taylor, finally solved the Fermat Theorem. It had been one of the most famous problems in mathematics and Sophie Germain’s efforts made an important contribution to the discovery of a proof. Dr. Pengelley’s work is of interest to historians not only for the way he used primary sources to teach mathematical concepts but he also revived interest in an under-appreciated figure, Sophie Germain, whose achievements deserve to be more widely celebrated.