Sophie Germain's theorem explained

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 for odd prime p .

Formal statement

Specifically, Sophie Germain proved that at least one of the numbers

x , y , z must be divisible by p ^{2} if an auxiliary prime q can be found such that two conditions are satisfied:

1. No two nonzero

p ^{th} powers differ by one modulo q ; and p is itself not a p ^{th} power modulo q .

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

q 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]

References

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

Notes and References

1. Legendre AM . Adrien-Marie Legendre . 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. 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.