The diophantine equation x3 3 + y3 + z 3 − 2xyz = 0

We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from theory of binary cubic forms and the method of infinite descent, which are well understood in the purview of Elementary Number Theory(ENT). In the second theorem, we show, that the famous Fermat’s Last Theorem(FLT) for exponent 3 and the first theorem are equivalent. So Theorem1 and 2 constitute an alternate proof for the non-existence of integer solutions of this famous cubic Fermat’s equation. It is well known, that from L.Euler(1770) to F.J.Duarte(1944) many had given proof of FLT for exponent 3. But all proofs uses concepts, which are beyond the scope of ENT. Hence unlike other proofs the proof given here is as an ENT proof of FLT for exponent 3. We will denote the greatest common divisor of integers a1, a2, a3, ... by symbol (a1, a2, a3, ...). For the results from theory of Binary Cubic Forms, we follow L.J.Mordell[1]. Consider the binary cubic f(x, y) = ax + bxy + cxy + dy = {a, b, c, d}, with integer coefficients and discriminant D = −27ad + 18abcd + bc − 4ac − 4bd, where D 6= 0. The quadratic covariant H(x, y) is H(x, y) = (b − 3ac)x + (bc− 9ad)xy + (c − 3bd)y, = Ax +Bxy + Cy = {A,B,C} (1) with discriminant B2 − 4AC = −3D and the cubic covariant G(x, y) = −(27ad− 9abc+ 2b)x + 3(6ac + bc− 9abd)xy + 3(bc − 6bd+ 9acd)xy + (27ad − 9bcd+ 2c)y. Lemma1: f(x, y),H(x, y) and G(x, y) are algebraically related by the identity, G(x, y) + 27Df(x, y) = 4H(x, y) [1]. Lemma2: All integer solutions of X + 27kY 2 = 4Z, (X,Z) = 1 (2) are given by taking, X = G(x, y), k = D, Y = f(x, y), Z = H(x, y).