It is also possible to reduce to 9 [Mordell equations](https://en.wikipedia.org/wiki/Mordell_curve). 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](https://web.archive.org/web/20110214173301/http://tnt.math.se.tmu.ac.jp/simath/MORDELL/).