Rational points on $x^3 + y^3 = 2$

Question

Are there any rational points on $x^3 + y^3 = 2$ besides $(1,1)$? I guess that a proof that there are no more could go through mimicking a proof of Fermat's Last Theorem for $n=3$, but I did not succeed with the details with the elementary proof (i.e. without using $\mathbb{Z}[\omega]$).

I am also fine with a solution which uses very heavy theorems and/or justified computer computation (such as some special Sage function). Any help appreciated!

Answer 1

Consider its projective closure $C \subset \mathbb{P}^2$ given by $$X^3 + Y^3 - 2Z^3 = 0$$

Then $C$ together with the point $(1:-1:0)$ is an elliptic curve. In fact, one may transform this into Weierstrass form, and see that it is isomorphic over $\mathbb{Q}$ to the elliptic curve with Weierstrass equation $$y^2 = x^3 - 27$$ and Cremona label $36a3$.

This curve has torsion subgroup isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and rank $0$ (this may be seen from a $2$-descent as Magma does, or simply from the information on this curve in the LMFDB).

In particular we see that $C$ has exactly $2$ rational points - namely the obvious ones, $(1:-1:0)$ and $(1:1:1)$.

Here is Magma code to check this

P<X,Y,Z> := ProjectiveSpace(Rationals(), 2);
E := Curve(P, X^3 + Y^3 - 2*Z^3);
E := EllipticCurve(E, E![1,-1,0]); 

MinimalModel(E);
Rank(E);
TorsionSubgroup(E);
CremonaReference(E);