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:
# No two nonzero $p^$ powers differ by one modulo $q$; and
# $p$ is itself not a $p^$ 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. Didot, Paris, 1827. Also appeared as Second Supplément (1825) to ''Essai sur la théorie des nombres'', 2nd edn., Paris, 1808; also reprinted in ''Sphinx-Oedipe'' 4 (1909), 97–128.

Notes

References

* Laubenbacher R, Pengelley D (2007
"Voici ce que j'ai trouvé": Sophie Germain's grand plan to prove Fermat's Last Theorem
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* {{cite book , author = Ribenboim P , author-link = Paulo Ribenboim , year = 1979 , title = 13 Lectures on Fermat's Last Theorem , publisher = Springer-Verlag , location = New York , isbn = 978-0-387-90432-0 , pages = 54–63

Theorems in number theory
Fermat's Last Theorem