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When $k=1$ this is trivial, and when $k=2$ the solutions correspond to the nonzero integral points on the elliptic curve:

$y^2=\frac{x(x+1)(2x+1)}{6}.$

And the Wolfram|Alpha says there are only four nonzero integral points on it ($x=1$ and $x=24$), hence two solutions. I am looking for an (algebro-geometric) proof for this fact, thanks! Also, are there any general results known for the higher $k$ cases?

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    I suggest this question be migrated to mathoverflow. It does seem like it is research level.2012-06-10

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