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 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 $p^{\mathrm {th} }$ powers differ by one modulo $q$; and
2. $p$ is itself not a $p^{\mathrm {th} }$ 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.

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.