What is the largest prime one less than a cubic number?

Consider the largest prime $p$ such that $p+1 = x^3$.

We can phrase this as $p = x^3 - 1$ and factor.
$$
p = (x^2 + x + 1)(x - 1)
$$

For $p$ to be prime either $x^2 + x + 1 = \pm1$
 or $x - 1 = \pm1$. 
 We can see $x$ cannot be $0$ or $-1$, so we know that $x - 1 = 1$.
$$
x = 2
$$

Thus the *only* prime one less than a cubic number is $7$.

Going further, for any prime $p$ followed by a number of the form $x^n$, $x$ must be $2$.

Edit: [amluto on HN](https://news.ycombinator.com/item?id=17904921) pointed out that this proof should consider
 negative values for factors