Question about parametric representations of solutions to $x^3+y^3+z^3=n \in \mathbb N$

Question

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 $n=1,2$?

Answer 1

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