Without loss of generality, let X <= Y. Then X! divides the left hand side, and also X! and Y!, so it must divide Z^2.
Dividing by X! gives
Y! = 1 + Y!/X! + Z^2/X!
let p be the smallest (first) prime > X/2
Consider each of the sides of the equation, mod p.
Y! (mod p) = 0
A little less obviously, Z^2/X! (mod p) = 0.
This is because z^2 must be a multiple of p^2, while x! is a multiple of p (but not p^2)
Therefore Y!/X! cannot be a multiple of p, because the equation mod p become 0 = 1 + 0 + 0. Therefore, Y < 2p.
Also, Y cannot equal X, because the equation mod p becomes 0 = 1 + 1 + 0.
This is possible if p = 2, but this leads to the solution (2,2,0) which is not allowed because Z is not positive.
Thus, Y must > X but very close to X.
For instance, if X = 2, then p = 2 and Y can only be 3. (this leads to solution (2,3,2)
For instance, if X = 3, then p = 2 and no Y is possible.
For instance, if X = 4, then p = 3 and Y can only be 5.
...
For instance, if X = 31, then p = 17 and Y can only be 32 or 33.
For instance, if X = 32, then p = 17 and Y can only be 33.
For instance, if X = 33, then p = 17 and no Y is possible.
In fact, this is the case whenever X is 1 more than twice a prime.
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If X = 10,000, then p = 5003 and Y can only be between 10001 and 10005
That's as far as I've gotten, but I suspect a full proof is available by heading in this direction.
Some analytical conclusions
