Proof verification: finding all prime numbers in the form of $n^3-1, n>1$

Question

Let $p$ be a prime number of the form $p = n ^3 - 1$ for a positive integer $n \geq 2$.

Then, factoring the difference of perfect cubes, we obtain $p = (n-1)(n^2 + n + 1)$.

Since $p = 1 \cdot p$ as well, and $n^2 + n + 1 > 1$, $n$ must satisfy $n-1=1$, thus implying $n=2$, yielding $p=7$.

Is this proof valid? Is $p=7$ the only prime number of this form?

Answer 1

As the others said in the comments: Yes, your proof is valid.