Modular Arithmetic/Sophie Germain's Theorem

Sophie Germain’s Theorem

Let $p$ be a prime number. Then, for the equation,

$x^{p}+y^{p}=z^{p}$

it is true that at least one of the numbers $x,y,z$ must be divisible by $p^{2}$, if and only if there exists a prime $q$, such that:

1. No two nonzero $p^{th}$ powers differ by one modulo $q$;
2. $p$ is itself not a $p^{th}$ power modulo $q$.

Corollary: The first case of Fermat's Last Theorem (the case in which $p$ does not divide $xyz$) must hold for every prime $p$ if there exists a prime $q$ for which (1) and (2) hold.