If you are not familiar with Eisenstein Integers, please start here. The details of today's blog are based on an English translation of Heinrich Dorrie's 100 Great Problems of Elementary Mathematics.
For those who would like to see this proof done in terms of rational integers only, check out this previous blog.
As in previous blogs, I will use Greek letters to represent quadratic integers and Latin letters to represent rational integers.
Theorem: The equation α3 + β3 = γ3 does not have any integer solutions where
α, β, γ, are Eisenstein integers and α * β * γ ≠ 0.
(1) Since Eisenstein Integers are Euclidean (proof), we know that they are characterized by a Division Algorithm (proof), Bezout's Identity (proof), and Unique Factorization (proof).
(2) We can assume that α, β and γ are coprime. [See here for proof]
(3) If we set ζ = -γ, then we have:
α3 + β3 + ζ3 = 0.
J = (1 + i√3)/2
O = (1 - i√3)/2
(5) Then, J - O is an Eisenstein prime number (proof ) which divides α*β*ζ [See here for proof.]
(6) But if α * β * γ ≠ 0, then we have an infinite descent. [See here for proof.]