In Hardy & Wright "An Introduction to the Theory of Numbers" there are two theorems:
Theorem 233: There are positive rationals which are not sums of two non-negative rational cubes.
Theorem 234: Any positive rational is the sum of three positive rational cubes.
The first one is proven by providing a counterexample - the number $3 \in \mathbb Q$, the second one is constructively proven using elementary number theory.
Now I wondered, can we classify the rationals $r \in \mathbb Q$ that satisfy theorem $233$ - the rationals that are not sums of one or two (but three) positive cubes?
You have run into a problem of probable not total solution until the end of times. Actually, you want to know for which rational numbers $A$ (you can assume without loss of generality that $A$ is positive integer) the equation $X ^ 3 + Y ^ 3 = AZ ^ 3$ has rational solutions.
This equation represents an elliptic curve from which the first one who studied it closely was the Norwegian mathematician E. S. Selmer (1920-2006) who calculated (without computers!) a very laborious table from $1$ to $166$ in which the integers representable by this equation ($6,7,9,37,61,….$) appear and in which implicitly the integers that do not appear are those that cannot be represented ($10,11,21,54,55,56,…..$).