The Mordell curve $$y^2=x^3+k$$ is known to have finite many solutions for every integer $k\ne 0$. But is it known whether there are infinite many natural numbers $n$ such that $y^2=x^3-n!$ and/or $y^2=x^3+n!$ has an integral point ? In other words :
Have the equations $$y^2=x^3+n!$$ and/or $$y^2=x^3-n!$$ infinitely many integral solutions ?
I checked the "+"-version for $0\le x\le 10^8$ and $1\le n\le 50$ and found the following squares
? for(s=1,50,z=s!;for(n=0,10^8,if(issquare(n^3+z)==1,print(s," ",n))))
1 0
1 2
4 1
4 10
4 8158
5 1
6 4
7 1
9 9
15 54180
21 604800
21 2419200
21 7358400
21 9676800
21 16805376
21 25363584
21 67536000
?
For small values $n$, the possible values can be looked up at tables, but for larger $n$ I am not sure how much is known about possible solutions.