Systematic way to find the answers of such question Q.What is the largest perfect square that divides:
$2014^3-2013^3+2012^3-2011^3+\ldots+2^3-1^3$.
My efforts:

$2014^3-2013^3=(2014-2013)(2014^2+2013^2+2014 \cdot2013)=2014^2+2013^2+2014*2013$
lly,
$2014^3-2013^3+2012^3-2011^3+\ldots+2^3-1^3=(2014^2+2013^2+2012^2+2011^2+\ldots+2^2+1^2)+(2014 \cdot2013+2013 \cdot2012\ldots+2 \cdot1)$

This is inconclusive.
So I am  looking for systematic way to find solutions of such problem.Any hint is appreciated.

$$ \sum_{k=1}^n \left((2k)^3 - (2k-1)^3\right) = \sum_{k=1}^n \left(12 k^2 - 6 k + 1\right) = (4n+3) n^2$$
In this case $n=1007$ and $4n+3 = 29 \times 139$ which is coprime to $n$ and squarefree.  Thus the answer is $n^2 = 1007^2 = 1014049$.
