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 ${\displaystyle p}$.

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:

1. No two nonzero ${\displaystyle p^{\mathrm {th} }}$ powers differ by one modulo ${\displaystyle q}$; and
2. ${\displaystyle p}$ 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

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.