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.
Lemma: α3 + β3 + γ3 = 0 → that J-O divides one and only one of α, β, or γ.
(1) Assume that J-O doesn't divide any of them.
(2) We know that there exists e,f,g such that:
α3 ≡ e (mod 9)
β3 ≡ f (mod 9)
γ3 ≡ g (mod 9)
(3) e2 = f2 = g2 = 1. [See here for proof]
(4) But then e + f + g is not ≡ 0 (mod 9) since: [See here for a review of modular arithmetic]
Case I: 1,1,1 : e + f + g ≡ 3 (mod 9)
Case II: -1,1,1 : e + f + g ≡ 1 (mod 9)
Case III: -1,-1,1 : e + f + g ≡ -1 (mod 9)
Case IV: -1,-1,-1 : e + f + g ≡ -3 (mod 9)
(5) And yet this contradicts α3 + β3 + γ3 ≡ 0 (mod 9).
(6) So we reject our assumption at step 1.
QED
Corrolary: Only 1 of the three values is divisible by J - O
(1) We know this since α, β, γ are relatively prime to each other.