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 $p^{\mathrm {th} }$ powers differ by one modulo $q$ ; and
$p$ is itself not a $p^{\mathrm {th} }$ 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.^[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.