If $ p \equiv 2 \mod 3$, make a conjecture as to which $a_i's$ are cubic residues.

Question

If $ p \equiv 2 \mod 3$, make a conjecture as to which $a_i's$ are cubic residues. Prove your conjecture.

Answer 1

Every $a$ relatively prime to $p$ is a cubic residue of $p$.

For by Fermat's Theorem we have $$a\equiv a^p \equiv a^{2p-1}\pmod{p}.$$ But since $p\equiv 2\pmod{3}$, it follows that $2p-1$ is divisible by $3$. Let $k=(2p-1)/3$ and $b=a^k$. Then $b^3=b^{2p-1}\equiv a\pmod{p}$.

Answer 2

Since $p-1\equiv 1 \bmod 3$, we have $\gcd(p-1,3)=1$ and so the map $x \mapsto x^3$ is injective and hence a bijection. So, every element is a cube.