Sophie Germain: mathématicienne extraordinaire - Scripps College

Chapter 5
Sophie Germain
5.1 Case 1, Case 2
Fermat’s Last Theorem states that the equation x n + y n = z n has no nontrivial
integer solutions when n > 2. Recall we may assume that x, y, and
z are relatively prime. Earlier, for n = 5, we broke this theorem down into
two cases. We can do this in general in the following manner:
Fermat’s Last Theorem 4 (By Cases).
Case 1. We claim that the equation x n + y n = z n has no non-trivial integer solutions
when n > 2 and n ∤ xyz.
In other words, if there did exist any positive integer solution (a, b, c) of
x n + y n = z n , then exactly one of a, b, or c is a multiple of n, since a, b, and
c are relatively prime.
Case 2. We claim that the equation x n + y n = z n has no non-trivial integer solutions
when n > 2 and n|xyz.