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?



Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$
We can get $N=6$ from $1+5+5=2+3+6$, $1^3+5^3+5^3=2^3+3^3+6^3$. 


We can get $N=5$ from $2+4+10=7+9$, $2^3+4^3+10^3=7^3+9^3$. 


