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.

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.

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

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.