TITLE: Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?
QUESTION [14 upvotes]: It's hard to do a Google search on this problem.
If I was using Maple correctly, there are no other positive solutions with n at most 10000.
I know some of these Diophantine questions succumb to known methods, and others are extremely difficult to answer.

REPLY [3 votes]: I realise this is a massive revival/refresh of this question, but I just found a completely elementary solution of this problem, outlined in this MSE question and my own answer.
Does anyone know if there are other completely elementary solutions?