Cubic Residues (Number Theory) I'm trying to figure out a conjecture about cubic residues and I need some examples
Any examples $a,b$ so that neither $a$ or $b$ is a cubic residue $\mod 19$, but $ab$ is a cubic residue modulo $19$?
Any examples where none of $a, b$ or $ab$ are cubic residues?
 A: Neither $2^6$ or $2^9$ are $\equiv 1\pmod{19}$, hence it follows that $g=2$ is a generator for $\mathbb{F}_{19}^*$ and:
$$\left\{ g^\color{purple}{0},g^3,g^6,g^9,g^{12},g^{15}\right\} = \color{purple}{\{1,8,7,18,11,12\}}, $$
$$\left\{ g^\color{red}{1},g^4,g^7,g^{10},g^{13},g^{16}\right\} = \color{red}{\{{2},16,14,17,3,5\}}, $$
$$\left\{ g^\color{blue}{2},g^5,g^8,g^{11},g^{14},g^{17}\right\} = \color{blue}{\{4,{13},9,15,6,10\}}, $$
so, for instance, neither $2$ or $13$ are cubic residues, but:
$$ \color{red}{2}\cdot\color{blue}{13}\equiv \color{purple}{7}\pmod{19}$$
and $7$ is a cubic residue. More generally, the multiplication table of colours is the following:
$$\begin{array}{c|c|c|c|}\cdot &\color{purple}{\bigstar} &\color{red}{\bigstar} & \color{blue}{\bigstar} \\ \hline \color{purple}{\bigstar} &\color{purple}{\bigstar} &\color{red}{\bigstar} &\color{blue}{\bigstar} \\ \hline \color{red}{\bigstar} &\color{red}{\bigstar} &\color{blue}{\bigstar} &\color{purple}{\bigstar} \\ \hline \color{blue}{\bigstar} &\color{blue}{\bigstar} &\color{purple}{\bigstar} &\color{red}{\bigstar}\\ \hline\end{array}$$
and the colour group, not by chance, is isomorphic to $\mathbb{Z}/(3\mathbb{Z})$.
A: take any prime $p$ that is congruent to $1\bmod 3$. The multiplicative group $\bmod p$ is isomorphic to the group $\mathbb Z_{p-1}=\{e,a,1^{1},a^2\dots a^{p-2}\}$ and hence the cubic residues are the elements with exponents that are multiples of three. So for example $a^1\cdot a^4=a^5$ wont be a cubic residue. And any non-cubic residue squared won't be a cubic residue.
To give an example consider $\bmod 19$,we have that $10$ is a primitive root.
We have $10\cdot10^4\equiv 10\cdot 6\equiv 3$ which is not a cubic residue.
We have $10\cdot 10\equiv 5$ which is not a cubic resiude.
A: The cubic residues $\pmod {19}$ are ${1, 7, 8, 11, 12, 18}$. See Jack D'Aurizio's answer, the product of a single "red" residue with a "blue" residue will yield a cubic residue. In particular for all primes $p=1 \pmod 3$ but not $p=1 \pmod 9$, if $c$ is an element with order $(p-1)/3 \pmod p$ (and therefore a cubic residue), and $r$ is a solution to $r^2+r+1=0 \pmod p$, then $ab$ is a cubic residue $\pmod p$ if and only if $a=rs^2$ for some integer $s$ and $b=r^2t^2$ for some integer $t$. 
