Sophie Germain and Special Cases of Fermat's Last Theorem Colleen Alkalay-Houlihan 1 Sophie Germain 1.1 A Passion for Mathematics Marie-Sophie Germain was a self-taught mathematician with an exceptional desire to learn mathe- matics. Born April 1, 1776 in Paris, France, revolution surrounded Germain's childhood. She came from a relatively prosperous family; her father was a silk merchant and was elected a member of the Constituent Assembly in 1789, when Germain was 13 years old. Escaping the chaos of revolution, Germain retreated into her father's library, where she discovered mathematics. The first books to enrapture her were Etienne B´ezout'sstandard mathematical textbooks and Jean-Etienne Montucla's \Histoire des math´ematiques."[2]Here she read the legend telling of how Archimedes, circa 212 B.C., was examining a geometric figure in the sand with such concentration that he did not respond to an enemy soldier's questions, and was speared to death. Germain began to realize that mathematics must be a fascinating subject, to inspire such absolute focus in a person, and henceforth she dedicated her time to studying math from the books in her father's study. In this time, however, women were not known to study mathematics; some aristocratic women were expected to have a basic understanding, but just enough to sustain a social conversation.[5] Germain, on the other hand, studied late into the night, absorbed in her attempts to understand, unguided, B´ezout'stexts. As her friend Guglielmo Libri recounted in her obituary, Germain's parents disapproved of her mathematical obsession. For this reason, they put out the fire in her room at night, and also took away her candles and clothing after nightfall, so that she would be unable to study. Even this did not deter her. She would study \at night in a room so cold that the ink often froze in its well, working enveloped with covers by the light of a lamp."[2] In 1794, when Germain was 22, the Ecole Polytechnique was founded in Paris, with the purpose of training exceptional young mathematicians and scientists. It was open only to male students, and so Germain was unable to attend classes, given by mathematicians such as Joseph-Louis Lagrange. However, she was able to obtain weekly lecture notes by using the alias \Antoine-August Le Blanc," and she completed and submitted assignments under this same name. Her answers to these problem sets were \ingenious," and she also submitted written \observations" that so impressed Lagrange that he expressed an interest in meeting his student. Germain was forced to reveal her true identity, and Lagrange became her friend and mathematical mentor. Amazingly, she had taught her self mathematics all the way to the undergraduate level, and despite this weak background, her work was impressive enough to merit the praise of one of the foremost mathematicians of her time.[5, 3] 1.2 Sophie Germain's Interest in Number Theory Germain became interested in number theory after reading Adrien-Marie Legendre's Essai sur la th´eoriedes nombres (1789) and Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801).[1] She 1 began work on Fermat's Last Theorem, which I will discuss in this text. She worked on original research for several years, until she had accomplished and understood a great deal, and she decided she needed to discuss her work with a number theorist. Boldly, she wrote directly to Gauss, again using her pseudonym, sharing her new, more general approach to proving Fermat's Last Theorem. Gauss was impressed by her progress, and he began to correspond with her.[5, 1] Upon learning that his correspondent was not, in fact, M. Le Blanc, but rather Sophie Germain, a woman, Gauss responded, \But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds never- theless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius."[5] After her correspondence with Gauss petered off, and she no longer was supported in her mathematical endeavours, Germain turned to the study of physics. Sophie Germain died June 27, 1831 in Paris.[1] On her death certificate, she was recorded as having been a \renti`ere,"not a mathematician.[5] 2 A Special Case of Fermat's Last Theorem 2.1 An Introduction to Fermat's Last Theorem Originally posed by Pierre de Fermat in the late 1630's, Fermat's Last Theorem states that xn + yn = zn has no positive integer solutions for all n > 2. The original Latin statement, famously written by Fermat in the margin of his copy of Diophantus' Arithmetica, was \Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."(As quoted in [6]) This translates to \It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree. I have discovered a truly remarkable proof which this margin is too small to contain."[4] The proof of this mathematical conjecture, deceptively easy to state, eluded mathematicians for over three hundred and fifty years. Finally, in 1995, Andrew Wiles published a proof of a conjecture which had been previously shown to imply Fermat's Last Theorem. (This also relied on a related paper co-authored with his former doctoral student, Richard Taylor.) 2 2.2 Past Work on Fermat's Last Theorem Countless mathematicians have worked on Fermat's Last Theorem (FLT), including Euler, Leg- endre, Gauss, Abel, Dirichlet, Kummer, and Cauchy. Germain was in fact on of the first people to have a \grand plan" for proving the theorem for all primes p, rather than a more patchwork attempt to prove special cases.[4] Fermat himself proved the case n = 4. Using his method of infinite descent, he showed that the area of a right triangle with rational sides cannot be a perfect square. This is equivalent to the statement that there are no integer solutions to x4 + y4 = z2, which then implies that there are no 4 4 4 4 4 4 solutions to x + y = z . If there were integer solutions a; b; c 2 Z such that a + b = c , then letting d = c2 would give a4 + b4 = d2, which is a contradiction. Any integer > 2 is either a multiple of 4 or has an odd prime factor. Thus, it now suffices to prove FLT for all odd prime exponents. If n = pm where p is a prime, then xn + yn = zn has a solution implies (xm)p + (ym)p = (zm)p, which gives a solution for the exponent p. In 1770, Euler published a proof for the case p = 3, although some subtle points were not rigorously justified in his proof. At the time Germain began working on FLT, only the proofs for the cases n = 3 and n = 4 were known.[4] 2.3 Sophie Germain's \Grand Plan" In around 1816, the Academy of Sciences in Paris announced a competition with a prize for a proof of Fermat's Last Theorem. Germain neither submitted to this competition nor published any work on FLT, despite the fact that she worked a great deal on the problem and made notable progress. Due to the unpublished nature of her manuscripts, the extent of her work on FLT was not known until very recently. (See [3] for a detailed examination of her original manuscripts and her various efforts relating to FLT.) Germain's grand plan did not work out as desired, but she proved some other interesting results along the way. Historically, she has not been given credit for many of her results, some of which were later proven by Legendre, presumably independently.[3] In a letter to Gauss, dated May 12, 1819, she explained her idea for a general proof. She aimed to show that for all odd primes p, there are infinitely many auxiliary primes of the form p 2np + 1, where n 2 Z≥1, such that the set of non-zero residues x mod (2np + 1) for 1 ≤ x ≤ 2np does not contain any consecutive integer residue classes mod (2np + 1). For example, if ap ≡ m mod (2np + 1) and bp ≡ (m + 1) mod (2np + 1) for some a; b 6= 0, then this condition would not be satisfied, since m and m + 1 belong to consecutive integer residue classes. She then showed that given an integer solution to xp + yp = zp, all auxiliary primes satisfying the non-consecutive pth power residue condition necessarily divide either x; y; or z. I will prove this second statement after a brief example of auxiliary primes. Germain analysed the auxiliary primes with 1 ≤ n ≤ 10 for all primes less than 100. What is important about Germain's technique is that she was the first person we know of who attempted to prove FLT for infinitely many prime exponents, rather than just on a case-by-case basis.[4] This philosophy reflects modern attempts to prove the theorem for infinitely many primes at once.