Sophie Germain: mathématicienne extraordinaire - Scripps College

Sophie Germain’s Theorems 61
if none of x, y, or z are congruent to 0 (mod q), then
x p ≡ ±1 (mod q)
y p ≡ ±1 (mod q)
z p ≡ ±1 (mod q).
This tells us that x p + y p + z p ≡ ±1 ± 1 ± 1 (mod q). This sum can never
be 0 (mod q). Thus if q ∤ xyz, then x p + y p + z p ≢ 0 (mod q). Condition 2 of
Sophie Germain’s theorem holds. □
Legendre published his version of the proof of Sophie Germain’s Theorem
in 1879. His mémoir expands the list of primes to include the 43
odd primes p up to 193.
In this publication, he proved Case 1 of Fermat’s
Last Theorem for primes p such that q = 2kp + 1 is also prime where
k ∈ {1, 2, 4, 5, 7, 8}. See Table 1.