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2020 | OriginalPaper | Chapter

9. Germain and Fermat’s Last Theorem

Author : Dora Musielak

Published in: Sophie Germain

Publisher: Springer International Publishing

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Abstract

On the same day Sophie Germain won the prize for her mathematical work on vibrating plates, the commissioners of the Academy of Sciences announced the (last) theorem of Fermat as the topic for the 1818 contest of mathematics. A happy coincidence? Germain kept her feelings secret, or at least there are no written words to shed light on her intellectual delight, as the new competition would carry her back to her first love—number theory. It would give her a fresh impetus to pursue once again the proof of the célèbre équation de Fermat, as she called it, an effort she had begun years earlier.

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previous chapter Elasticity Theory After Germain

next chapter Pensées de Germain

www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html.

Edwards (1977), p. 8.

Pierre de Carcavy (1600/1603–1684). Mathematician, secretary of the royal library of Louis XIV.

Bernard Frénicle de Bessy died in Paris in 1674.

Letter from Fermat to Frenicle dated August 1640. Translated by Amanda Bergeron and David Zhao. Available at The Euler Archive.

Ibid.

Fermat’s little theorem is the basis for a primality test, a check to determine if a number is prime or composite.

Ribenboim (1979), p. 2.

Oeuvres de Fermat. Bibliothèque Nationale de France. http://gallica.bnf.fr.

Fuss (1843), Letter II dated 1 December 1729, p. 10.

Ibid. Letter III dated 8 January 1730, p. 18. [Sed nondum prorsus persuasus sum, quomodo sola inductione id inferre legitime potuerit, cum certus sim ipsum numeris in formula \( 2^{{2^{x} }} \) loco \( x \) substituendis nec ad senarium quidem pervenisse.]

Matvievskaya and Ozhigova (2007), p. 131.

Euler (1732).

Sandifer (2003).

Sandifer (2007), p. 284.

Euler (1738).

Sandifer (2007), p. 287.

Euler (1770).

Fuss (1842), p. 618.

Legendre (1798), p. 410. He presented a copy to the Institut on 13 July 1798.

Institut de France. Procès-verbaux. Tome V, p. 592.

Ibid., p. 596.

Ribenboim (1979), p. 3.

Institut de France. Procès-verbaux. Tome VI, p. 242. The motto is a phrase adapted from Virgil’s Georgics translated as “Work conquers all.”.

Institut de France. Procès-verbaux. Tome VI, pp. 242, 262.

Généralisation du théorème de Fermat sur les doubles égalités. Institut de France. Procès-verbaux. Tome VI, p. 418.

Schumacher wrote: “Verzeihen Sie die Eile dieses Briefes. Ich bin glücklich zu Mlle. Germain durchgedrungen und soll Mittwochen bei ihr essen. Das nähere mündlich auf meinem Durchfluge durch Göttingen, wo ich leider nur einige Augenblicke bleiben kann.” Peters, C.A.F., Ed. 1860. Briefwechsel zwischen C.F. Gauss und H.C. Schumacher, erster Band. Altona, Esch. p. 159.

From Lactantius: “Those aspiring to perfection, must arrive there through the most unpleasant difficulties.”

From Cicero’s Orator: “I will take it on, not so much out of hope of completing it as out of willingness to try.”

Institut de France. Procès-verbaux. Tome VI, pp. 507, 521; Tome VII, p. 8.

Papiers de Sophie GERMAIN. Français 9115. A collection of Germain’s handwritten notes showing her research and analysis.

The Institute split the 3000 francs between French astronomer M.-C. Damoiseau, and Italian astronomers F. Carlini and G. Plana, who developed new Lunar Tables; see Institut de France. Procès-verbaux. Tome VII, p. 18.

Institut de France. Procès-verbaux. Tome VIII, p. 241.

Legendre (1825).

Dirichlet (1825). The Addition with the final proof was attached to page 12 of the original memoir.

Legendre (1825), pp. 13–14.

For a proof of Sophie Germain’s theorem, see Ribenboim (1999), chap. IV.

Del Centina (2008), pp. 349–392.

Laubenbacher and Pengelley (2010).

Ibid., pp. 641–92.

Del Centina (2008), p. 362.

Del Centina (2005), p. 62.

Poinsot (1817), p. 381.

Del Centina (2008), p. 361.

Stupuy (1896), pp. 310–311.

Legendre (1830).

Laubenbacher and Pengelley (2010), pp. 641–92.

Ibid.

Ibid., pp. 657, 682.

Ibid., p. 679.

Ibid., p. 662.

Institute de France. Procès-verbaux. Tome VIII, pp. 121–123.

Ibid.

Ibid., p. 223.

Del Centina (2005), p. 8.

Institut de France. Procès-verbaux. Tome VIII, p. 358.

Mémoires de l’Académie des sciences de l’Institut de France (1830). Tome IX, Année 1826. Chez Firmin Didot Frères, libraries, Paris. pp. xix–xxx.

Edwards (1977), p. 61.

Ibid., p. 60.

Ribenboim (1999), Appendix A.

Wiles (1995).

Taylor and Wiles (1995).

http://mathworld.wolfram.com/ModularForm.html.

For a treatment of number theory where these concepts are expounded, see for example, Jean-Pierre Serre (1973).

Davenport (2008), p. 202.

Havil (2010), p. 163.

Title: Germain and Fermat’s Last Theorem
Author: Dora Musielak
Publisher: Springer International Publishing
Book: Sophie Germain
Print ISBN: 978-3-030-38374-9

Electronic ISBN: 978-3-030-38375-6

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-38375-6_9

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