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.

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]

• 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.