It is also possible to reduce to 9 Mordell equations. Denoting $x := 2^{[i/3]}3^{[k/3]}$, we get 9 equations: $$d x^3 + 1 = y^2$$ indexed by $d\mid 2^23^2$.
Equivalently, $$(dy)^2 = (dx)^3 + d^2,$$ which are Mordell equations with known solutions.
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Max Alekseyev
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Max Alekseyev
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