7
$\begingroup$

There are such representations for $n=1,2$. However, by the Wikipedia article, it seems that there are no known parametric (polynomial) representations $P,Q,R$ such that $(P(m))^3+(Q(m))^3+(R(m))^3=3$.

Is there any $n \in \mathbb N$ (other than those trivially excluded with conguence conditions) for which it is known that there do not exist three polynomials $A,B,C$ with integer coefficients such that $(A(d))^3+(B(d))^3+(C(d))^3=n$. And are there some general conjectures about non-representability of some natural numbers? Do some heuristics suggest that it is rare to have such a representation, as we have for $n=1,2$?

$\endgroup$
2
$\begingroup$

Above equation shown below:

$x^3+y^3+z^3=n$

There is a parametric solution for rational values of

$(x,y,z)$ and $(n= 3)$ given by Seji Tomita & is shown below.

$x=w(27a^3+9ab^2-4b^3)$

$y=-w(27a^3+9ab^2+4b^3)$

$z=wb(27a^2+5b^2)$

$w=[(1)/((b(3a+b)(3a-b))]$

For, $(a,b)=(1,2)$ we have,

$(x,y,z)=(31/10,-95/10, 94/10)$

The link to his site is given below. Click on

"Computational number theory" & select article #104.

http://www.maroon.dti.ne.jp/fermat

$\endgroup$
  • 4
    $\begingroup$ This is a parameterization by rational functions in $a$ and $b$, but the question asks for polynomials (with integral coefficients). $\endgroup$ – Michael Stoll Jun 16 at 8:55

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.