Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation xp + yp = zp of Fermat's Last Theorem. Specifically, Sophie Germain proved that the product xyz must be divisible by p2 if an auxiliary prime can be found such that two conditions are satisfied: 1. No two pth powers differ by one modulo ; and 2. p is itself not a pth power. Conversely, the first case of Fermat's Last Theorem must hold for every prime p for which even one auxiliary prime can be found. Germain identified such an auxiliary prime 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.

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