Number Theory or Algebra? Part 9

Find the number of possible co-ordinate \((x, y)\), in which \(x\) and \(y\) are both positive integers, which satisfies the equation

\(\sum^{1999}_{i = 1}i^{2538} = 4x^{2} + 8y^{3} + 2\)

Details and Assumptions :

\(i\) is a dummy variable, not \(\sqrt{-1}\)

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