Sophie Germain: mathématicienne extraordinaire - Scripps College

The First Proof 39
the previous 3 chapters.
4.3.2 The Problem
We will divide the proof of Fermat’s Last Theorem for n = 5 into two cases.
Recalling our standing assumption that x, y, and z are relatively prime, we
state the problem as follows:
Fermat’s Last Theorem 3 (for n=5).
Case 1. We want to prove that the equation X 5 + Y 5 + Z 5 = 0 has no non-trivial
integer solutions x, y, and z such that 5 ∤ xyz.
Case 2. We want to prove that the equation X 5 + Y 5 + Z 5 = 0 has no non-trivial
integer solutions x, y, and z such that 5|xyz.
In this case, since five divides the product xyz and since x, y, and z are
relatively prime, 5 divides exactly one of x, y, and z. We also know that
exactly one of x, y, and z is even as discussed above in the presentation of
Primitive Pythagorean Triples. Then either 5 and 2 divide the same number
or 5 and 2 divide different numbers. Thus we have two subcases of Case 2:
Case 2. A.
We want to prove that the equation X 5 + Y 5 + Z 5 = 0 has no
non-trivial integer solutions x, y, and z such that 5 divides one of x, y, or z
and that the number divisible by 5 is even. That is, 10 divides one of x, y,
or z.
Case 2. B.
We want to prove that the equation X 5 + Y 5 + Z 5 = 0 has no
non-trivial integer solutions x, y, and z such that 5 divides one of x, y, or z