TITLE: Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$
QUESTION [10 upvotes]: For $x_i \in \mathbb{Z}$, let  $\{x_i\}$ be a fundamental solution to the equations:
$$
\sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0
$$
if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$.
For instance, a fundamental solution with $N=7$ is given by
$$
x_1 = 4, \quad  x_2 = x_3 = x_4 = -3, \quad  x_5 = x_6 = 2, \quad  x_7 = 1
$$
What is the minimum $N$ for which a fundamental solution exists?

REPLY [3 votes]: If $N=6$ then we have a nice identity:
$$(x_1+x_2)(x_1+x_3)(x_2+x_3)=-(x_4+x_5)(x_4+x_6)(x_5+x_6).$$
(Loo-Keng Hua used such identities in his "Additive Theory of Prime Numbers".) In particular if $x_5=x_6=0$ then LHS vanishes. From this observation easily follows that $N>4$.