Originally Posted by
topsquark 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:
$\displaystyle x^3 + (x + 1)^3 + (x + 2)^3 +....+ (x + 7)^3 = y^3$
becomes (after a fair amount of work)
$\displaystyle 8x^3 + 84x^2 + 420x + 784 = y^3$
$\displaystyle 8x^3 + 84x^2 + 420x + (784 - y^3) = 0$
$\displaystyle x^3 + \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x + 98 - \frac{y^3}{8} = 0$
For this to be true for x and y integers we must have that
$\displaystyle \left ( \frac{21}{2} \right ) x^2 + \left ( \frac{105}{2} \right ) x - \frac{y^3}{8}$
must be an integer. That might be a good starting point.
-Dan