Random Diophantine Equation

Find the number of positive integers nn such that there exists an integer mn2m \neq \frac{n}{2} such that n32m36m\frac{n^3-2m^3}{6m} is a perfect square.


Details and Assumptions:

  • All perfect squares are integers. For example, 82,42,8^2, 4^2, etc. are perfect squares, but (23)2\left( \frac{2}{3} \right) ^2 isn't.
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