Sophie Germain: mathématicienne extraordinaire - Scripps College

Sophie Germain’s Theorems 59
fied and if there are integer solutions x, y, and z to the equation X p + Y p + Z p = 0,
then exactly one of x, y, or z is a multiple of p. This concludes our proof of
Sophie Germain’s Theorem. □
Sophie Germain found auxiliary primes q for all primes up 100, thus
proving Case 1 of Fermat’s Last Theorem for these 24 primes. She also
found that for primes p > 2 such that q = 2p + 1 is also prime, conditions
1 and 2 hold, as we will see below. These primes have come to be known
as Sophie Germain primes. Her theorem applies to all Sophie Germain
primes, thus proving Case 1 of Fermat’s Last Theorem for all odd primes p
such that q = 2p + 1 is also prime.
Sophie Germain Primes Theorem 2. Let p be an odd prime with the property
that q = 2p + 1 is also prime. Then conditions 1 and 2 hold. I.e.:
1. n p ≢ p (mod q) for all integers n.
2. x p +y p +z p ≡ 0 (mod q) implies that q|xyz where x, y, and z are relatively
prime.
Proof of the Sophie Germain Primes Theorem. Consider an integer n.
Clearly if n ≡ 0 (mod q), then, since q ∤ p,
n p ≡ 0 (mod q)
≢
p (mod q).
Fermat’s Little Theorem tells us that when q is an odd prime, n q−1 ≡ 1 (mod q)
for any integer n ≢ 0 (mod q).