<s>[INST] Find all solutions to $x^3+(x+1)^3+ \dots + (x+15)^3=y^3$ 
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!
Any ideas? 
 [/INST] You are looking for $(x, y) \in \mathbb{Z}^2$ for which 
$$ \sum_{i= 0}^{15} (x+i)^3 = y^3, \tag{0}$$
that is, 
$$ \sum_{i= 0}^{15} \left( x^3 + 3 i x^2 + 3i^2 x + i^3 \right) = y^3, $$
that is,
$$ 16 x^3 + 3 \frac{15 (15+1)}{2} x^2 + 3 \frac{15 (15+1)(2 \times 15 + 1)}{6} + \left( \frac{15 (15+1)}{2} \right)^2 = y^3, $$
that is, 
$$ 16 x^3 + 360 x^2 + 3720 x + 14400 = y^3, $$
which can be written as 
$$ 8 (2x^3  +  45 x^2 + 465 x + 1800)  = y^3  $$
Can you get anywhere from here?
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