# All integer is a sum of four cubes of rational

I am interested in a classic problem of representation of integers as a sum of four cubes of integers. This problem has been partially solved missing only integers of the form 9k ± 4. I got a proof that this is true for rational. This is not important really because it is not given in a ring but in a field (where the operability is easier). From this the following question:

Prove that all rational integer n is a sum of four cubes of nonzero rational numbers.

• What is a rational integer? – Wojowu Apr 21 '15 at 15:56
• Any element of Z, not only positive integers. Actually one can ask just for natural numbers because -1 is a cube. – Piquito Apr 21 '15 at 17:25
• @Wojowu A rational integer is an integer in $\mathbb{Q}$, in contrast to integers from other "rings of integers" $\mathcal{O}_K$ of a number field $K$. – Dietrich Burde Apr 21 '15 at 18:19

Every rational number is the sum of three rational cubes:

Theorem (Reyley 1825): Every rational number $r$ can be represented as the sum of three rational cubes: $$r=x^3+y^3+z^3.$$ A short proof can be found in the artcle of Richmond (1930).

For $a\in \mathbb{Q}$ we can represent $s-a^3=x^3+y^3+z^3$, so that $s=a^3+x^3+y^3+z^3$ for any rational $s$.

• Concerning the representation of an integer as a sum of r cubes, the value r = 3 of the paper you evoke is better than r = 4 and 3 is besides optimal. However it is not directly related, (I mean as a source) to the open problem on integers I have mentioned. Anyway, If you are motivated for, you could perhaps appreciate my solution based in just one parameter (neither two ones nor no real number as in the above paper) which is nothing less than the represented number itself. Informally, I get n = A + B + C +D where the uppercase letters are cubes, function of n. – Piquito Apr 21 '15 at 19:55
• Your substitution s - $a^3$ shows -unintentionally I guess- in a very elementary way what is inferred at first level of elliptic curves theory: the representations for each number n here are infinitely many (which also applies to three cubes of Ryley of course). – Piquito Apr 21 '15 at 20:25