x,y integers
x^3 +(x+1)^3 +(x+2)^3+....+(x+7)^3=y^3
There's quite a bit of work going on here! I would think that the simplest start to this would be to expand and simplify:
becomes (after a fair amount of work)
For this to be true for x and y integers we must have that
must be an integer. That might be a good starting point.
-Dan