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

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 $$n=1,2$$?

• – Will Jagy Jun 11 at 14:54

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

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