Sophie Germain's theorem

In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x^p + y^p = z^p of Fermat's Last Theorem.

Formal statement

Specifically, Sophie Germain proved that the product xyz must be divisible by p² if an auxiliary prime θ can be found such that two conditions are satisfied:

1. No two p^th powers differ by one modulo θ; and
2. p is itself not a p^th power modulo θ.

Conversely, the first case of Fermat's Last Theorem (the case in which p does not divide xyz) must hold for every prime p for which even one auxiliary prime can be found.

History

Germain identified such an auxiliary prime θ for every prime less than 100. The theorem and its application to primes p less than 100 were attributed to Germain by Adrien-Marie Legendre in 1823.^[1]

Notes

1. ↑ Legendre AM (1823). "Recherches sur quelques objets d’'analyse indéterminée et particulièrement sur le théorème de Fermat". Mém. Acad. Roy. des Sciences de l’'Institut de France 6.  C1 control character in |title= at position 33 (help); C1 control character in |journal= at position 34 (help) Didot, Paris, 1827. Also appeared as Second Supplément (1825) to Essai sur la théorie des nombres, 2nd edn., Paris, 1808; also reprinted in Sphinx-Oedipe 4 (1909), 97––128.

References

• Laubenbacher R, Pengelley D (2007) “Voici ce que j’ai trouvé:” Sophie Germain'’s grand plan to prove Fermat’'s Last Theorem
• Mordell LJ (1921). Three Lectures on Fermat's Last Theorem. Cambridge: Cambridge University Press. pp. 27–31. 
• Ribenboim P (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag. pp. 54–63. ISBN 978-0-387-90432-0.