Sophie Germain's theorem

Short description: On the divisibility of solutions to Fermat's Last Theorem for prime exponent

In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation [math]\displaystyle{ x^p + y^p = z^p }[/math] of Fermat's Last Theorem for odd prime [math]\displaystyle{ p }[/math].

Formal statement

Specifically, Sophie Germain proved that at least one of the numbers [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], [math]\displaystyle{ z }[/math] must be divisible by [math]\displaystyle{ p^2 }[/math] if an auxiliary prime [math]\displaystyle{ q }[/math] can be found such that two conditions are satisfied:

1. No two nonzero [math]\displaystyle{ p^{\mathrm{th}} }[/math] powers differ by one modulo [math]\displaystyle{ q }[/math]; and
2. [math]\displaystyle{ p }[/math] is itself not a [math]\displaystyle{ p^{\mathrm{th}} }[/math] power modulo [math]\displaystyle{ q }[/math].

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

History

Germain identified such an auxiliary prime [math]\displaystyle{ q }[/math] for every prime less than 100. The theorem and its application to primes [math]\displaystyle{ p }[/math] 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. https://archive.org/details/threelectureson00mordgoog. 
• Ribenboim P (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag. pp. 54–63. ISBN 978-0-387-90432-0.