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.