[math-fun] Brocard-like questions

Brocard's problem (Brocard 1876, 1885; Ramanujan 1913) is to decide whether 4! + 1 = 5^2 5! + 1 = 11^1 7! + 1 = 71^2 are the only factorial-plus-ones equal to a square: (*) n! + 1 = K^2 . It remains unsolved. Question: --------- What is known about generalizing to any exact power: n! + 1 = K^r (r >= 2) and what if we let +1 be replaced by -1 here or in (*) ??? Note: It was shown by Luca (2002) that the abc conjecture (https://en.wikipedia.org/wiki/Abc_conjecture) implies that at most finitely many solutions exist to the still more general equation n! = P(n) where P is any integer polynomial of degree at least 2. —Dan