the first case of Fermat's Last Theorem holds true for p. Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea Del Centina, "she had actually shown that it holds for every exponent p < 197". L. E. Dickson later used Germain's theorem to prove the first case of Fermat's Last Theorem for all odd primes less than 1700. In an unpublished manuscript titled , Germain showed that any counterexamples to Fermat's theorem for p > 5 must be numbers "whose size frightens the imagination", around 40 digits long. Germain did not publish this work. Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 (see Correspondence with Legendre). Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later. Del Centina states that "after almost two hundred years her ideas were still central", but ultimately her method did not work. Work in philosophy In addition to mathematics, Germain studied philosophy and psychology. She wanted to classify facts and generalize them into laws that could form a system of psychology and sociology, which were then just coming into existence. Her philosophy was highly praised by Auguste Comte. Two of her philosophical works, and , were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. is a history of science and mathematics with Germain's commentary. In , the work admired by Comte, Germain argues that there are no differences between the sciences and the humanities. Final years In 1829 Germain learned that she had breast cancer. Despite the pain, she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and "a note about finding and in ". Mary Gray records: "She also published in an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids." On 27 June 1831, she died in the house at 13 rue de Savoie. Despite Germain's intellectual achievements, her death certificate lists her as a "" (property holder), not a "". But her work was not unappreciated by everyone. When the matter of honorary degrees came up