# Sum of rational cubes

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?

• Since Theorem 233 refers to non-negative cubes, you intend: not sums of 1 or 2 positive cubes, correct? – coffeemath May 25 '18 at 19:32
• You're right, thanks for noticing! – flawr May 25 '18 at 19:36
• Theorem 234 might be Reyley's theorem (see here). – Watson Jul 17 '18 at 15:56

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 ($10,11,21,54,55,56,…..$).
• Yes: E. S. Selmer, The diophantine equation $ax^3+by^3+cz^3=0$, Acta Math. (Stockh.) $\mathbf{85}$ , p.203-362. – Piquito May 25 '18 at 21:00
• You are welcome. These curves have been well studied. In the tables you will also find the ranges corresponding to each case of $A$ what for Selmer must have been a very arduous job. It is also shown (after Selmer) that these curves (defined over an extension of rationals) have complex multiplication by $\sqrt{-3}$ (maybe you do not know this notion and I tell you with all purpose to see if you dare to know this topic later). – Piquito May 25 '18 at 21:17