Sophie Germain: mathématicienne extraordinaire - Scripps College

The First Attempts 33
4.2 The First Attempts
We now return to our investigation of possible solutions to the equation
x n + y n = z n when n > 2. Historically, the very first attempts at proving
Fermat’s Last Theorem were made on n = 3, 4, and 5.
4.2.1 n=3
Leonhard Euler (1707-1783) gave a proof in 1753 that the equation x 3 + y 3 = z 3
has no non-trivial integer solutions. Euler used a variation on the method
of infinite descent, which we will discuss in detail below in section 4.2.2.
Fifty years later, it was discovered that Euler’s proof was actually incomplete.
This incomplete proof was available to Sophie Germain; she and Legendre
worked on the case where n = 5 in a very similar fashion as shown
below in section 4.3.1. It is reported that Lagrange “finished” Euler’s proof
as a consequence of some of his work in the 1830’s. [6]
4.2.2 n=4
Fermat wrote his conjecture in the margin of his Oeuvres de Diophante; he
never furnished a proof. Later, however, he illustrated his method of infinite
descent specifically for n = 4.
Infinite Descent.
The process of infinite descent begins with an equation,
for example, x n + y n = z m , and assumes that x 0 , y 0 , and z 0 are positive
integer solutions. From the first solution, one derives a second solution of
the same form, x n 1 + yn 1 = zm 1
where x 1, y 1 , and z 1 are again positive integer