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.