Number Theory or Algebra? Part 9

If even integers can be written in the form of $e$ , and odd integers can be written in form of $o$ We will get: $e^{2538} \equiv 0 (mod 4)$

$o^{2538} \equiv 1 (mod 4)$

So, $\sum^{1999}_{i = 1}i^{2538} \equiv 1000 \equiv 0 (mod 4)$

But, $4x^{2} + 8y^{3} + 2 \equiv 2 (mod 4)$

So, $L. H. S. \not\equiv R. H. S.$ .

So, there's no chance of making both sides equal when $x$ and $y$ are both integers