# Sophie Germain's theorem

Sophie Germain's theorem In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation {displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime {displaystyle p} .

Contents 1 Formal statement 2 History 3 Notes 4 References Formal statement Specifically, Sophie Germain proved that at least one of the numbers {displaystyle x} , {displaystyle y} , {displaystyle z} must be divisible by {displaystyle p^{2}} if an auxiliary prime {displaystyle q} can be found such that two conditions are satisfied: No two nonzero {displaystyle p^{mathrm {th} }} powers differ by one modulo {displaystyle q} ; and {displaystyle p} is itself not a {displaystyle p^{mathrm {th} }} power modulo {displaystyle q} .

Conversely, the first case of Fermat's Last Theorem (the case in which {displaystyle p} does not divide {displaystyle xyz} ) must hold for every prime {displaystyle p} for which even one auxiliary prime can be found.

History Germain identified such an auxiliary prime {displaystyle q} for every prime less than 100. The theorem and its application to primes {displaystyle p} less than 100 were attributed to Germain by Adrien-Marie Legendre in 1823.[1] Notes ^ Legendre AM (1823). "Recherches sur quelques objets d'analyse indéterminée et particulièrement sur le théorème de Fermat". Mém. Acad. Roy. des Sciences de l'Institut de France. 6. 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. References Laubenbacher R, Pengelley D (2007) "Voici ce que j'ai trouvé": Sophie Germain's grand plan to prove Fermat's Last Theorem Mordell LJ (1921). Three Lectures on Fermat's Last Theorem. Cambridge: Cambridge University Press. pp. 27–31. Ribenboim P (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag. pp. 54–63. ISBN 978-0-387-90432-0. Categories: Theorems in number theoryFermat's Last Theorem

Si quieres conocer otros artículos parecidos a Sophie Germain's theorem puedes visitar la categoría Fermat's Last Theorem.

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