A Diophantine equation $$x^3 - Dy^3 = 1$$ always has a trivial solution $x = y^3 + 1$. It appears that a non-trivial (that is those with $x$ smaller than trivial) solution exists iff $y$ is a Heronian integer (that is $y = \frac{a + \sqrt{ab} + b}{3}$ for some integers $a \ne b$).
Obviously prime $y$s give the most interesting solutions. Which raises the actual question:
What is known about the Heronian primes?