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 for odd prime p.
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Formal statement
Specifically, Sophie Germain proved that at least one of the numbers x, y, z must be divisible by p2 if an auxiliary prime θ can be found such that two conditions are satisfied:
- No two nonzero pth powers differ by one modulo θ; and
- p is itself not a pth power modulo θ.
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 θ 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.
References
Sophie Germain's theorem Wikipedia(Text) CC BY-SA