Sophie Germain: mathématicienne extraordinaire - Scripps College

What then 64
p, 6p + 1” [9] explains why these pairs are exceptional and how to prove
Case 1 of Fermat’s Last Theorem for such primes.
What possibilities are left
In general, Sophie Germain’s Theorem precludes
most solutions x, y, and z to the equation X n +Y n = Z n where none
of the integers x, y, or z is divisible by n. We have already discussed the
impossibility of equations where n is divisible by 4. We have also seen that
when n is divisible by 2, any the squares of any solutions to the equation
X n +Y n = Z n are solutions to X n 2 +Y n 2
= Z n 2 . So we consider the non-zero
integer solutions to the equation
X P + Y P + Z P = 0
where P is a product of primes greater than 2.
We know that one of X, Y , or Z is divisible by P , without loss of generality
we assume X = aP for some integer a. Then (aP ) P + Y P + Z P = 0.
There are a few good number theoretic tools which help narrow down
possible solutions. We will consider only Fermat’s Little Theorem and the
Chinese Remainder Theorem, but there are other interesting approaches,
many of which are discussed at length in [23].
We recall Fermat’s Little Theorem,
a p ≡ a (mod )
for any integer a where p is prime.