I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$.
Note that this is a special case of Beal's conjecture, where in this case $A = p$, $B = n$, and $C = n + 1$ (and thus $A$, $B$, and $C$ do not have a prime factor).
(To those who are wondering, this is relevant to the classification of elliptic curves $y^2 = x^3 + B$ over $\mathbb{F}_{p^r}$ whose orders are $p^r$.)
Thank you!
