5
$\begingroup$

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?

|cite|improve this question
$\endgroup$
  • $\begingroup$ Since Theorem 233 refers to non-negative cubes, you intend: not sums of 1 or 2 positive cubes, correct? $\endgroup$ – coffeemath May 25 '18 at 19:32
  • $\begingroup$ You're right, thanks for noticing! $\endgroup$ – flawr May 25 '18 at 19:36
  • 1
    $\begingroup$ Theorem 234 might be Reyley's theorem (see here). $\endgroup$ – Watson Jul 17 '18 at 15:56
3
$\begingroup$

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 ($10,11,21,54,55,56,…..$).

|cite|improve this answer
$\endgroup$
  • $\begingroup$ Thanks a lot, it seems elliptic curves pop up everywhere:) Do you have a reference where you found those numbers? $\endgroup$ – flawr May 25 '18 at 20:51
  • 1
    $\begingroup$ Yes: E. S. Selmer, The diophantine equation $ax^3+by^3+cz^3=0$, Acta Math. (Stockh.) $\mathbf{85}$ , p.203-362. $\endgroup$ – Piquito May 25 '18 at 21:00
  • $\begingroup$ Great, thank you very much! $\endgroup$ – flawr May 25 '18 at 21:03
  • 1
    $\begingroup$ 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). $\endgroup$ – Piquito May 25 '18 at 21:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.