# Modular Arithmetic/Sophie Germain's Theorem

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**Sophie Germain’s Theorem**

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

it is true that at least one of the numbers must be divisible by , if and only if there exists a prime , such that:

- No two nonzero powers differ by one modulo ;
- is itself not a power modulo .

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