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.



Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?
sage: E = EllipticCurve([0,0,1,0,-1])


sage: E


Elliptic Curve defined by y^2 + y = x^3 - 1 over Rational Field


sage: E.integral\_points()


[(1 : 0 : 1), (7 : 18 : 1)]


