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$a, b$ are two naturals such that,

$a^{2}-b^{2}=k^{3}$ and $a^{3}-b^{3}=c^{2}$

where $k^{3}$ and $c^{2}$ are perfect cube and square respectively.

What can be the least possible pair of naturals $(a, b)$ for the above to hold true?

This link suggests $(10, 6)$ but I'm not satisfied with the answer. Please help.

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    Not sure what "analytically" means. You could have tried $(1,1)$, then $(1,2)$, then $(2,1)$, etc., etc., until you found one that works - the first one you found would be the answer.2012-08-09

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