Sophie Germain: mathématicienne extraordinaire - Scripps College

Sophie Germain’s Theorems 52
In other words, if there did exist any positive integer solution (a, b, c) of
x n + y n = z n , then none of a, b, or c are multiples of n.
5.2 Sophie Germain’s Theorems
These two cases developed following Sophie Germain’s Theorem:
Sophie Germain’s Theorem 1. Let p be an odd prime. Suppose there exists an
auxiliary prime q such that the following conditions hold:
1. n p ≢ p (mod q) for all integers n.
2. Whenever x p + y p + z p ≡ 0 (mod q) then q|xyz where x, y, and z are
relatively prime.
Then for any non-zero integers a, b, and c such that a p + b p = c p , it must be that
p|abc.
That is, suppose a prime p has an auxiliary prime q such that conditions
1 and 2 are met. If there exists non-zero integers a, b, and c for which
a p + b p = c p , then p divides a, b, or c.
Example.
Let us consider the primes p = 3 and q = 7. We check directly
conditions 1 and 2.
1. Is n 3 ≡ 3 (mod 7) for any integer n We can easily check this with a
table.