Find all pairs of integers $(x, y)$ such that $$x^3+(x+1)^3+ \dots + (x+15)^3=y^3$$
What I have tried so far:
The coefficient of $x^3$ is $16$ in the left hand side. It is not useful then to trying bound LHS between, for example, $(ax+b)^3$ and $(ax+c)^3$ and then say that $ax+b<y<ax+c$.
I also tried to use modulo a prime. But it seems unlikely to bound variables this way.
EDIT : Though, it can be factored as $(2x+15)(x^2+15x+120)=(y/2)^3$. LSH factors are almost co-prime and we can say that $x^2+15x+120=3z^3$ or $x^2+15x+120=5z^3$. These are still too difficult to solve!