The largest solution that I found to equation $x^2 + y^3 = z!$ with $y \ge 0$ as below.

$$6389296200^2 + 2173500^3 = 21!$$

It is obvious that if $z!/t^{6} = x_{t}^2 + y_{t}^3$ then $z! = (x_{t}t^3)^2 + (y_{t}t^2)^3$. So for a large value of $z$, there can be certain set of $t$ values to check existence of a smaller solution that provides $x^2 + y^3 = z!$ is soluble.

In that direction, I believe that there can be search to other solutions to this equation.

**Question.** Are there other solutions to $x^2 + y^3 = z!$ for $z > 21$? (It is checked up to $33!$).

I don't have any reference on that equation except https://oeis.org/A273553. So any reference is very welcome if this is well-known equation.

Thanks.