Am thinking that given n! + 1 = k^r Then for all integers given r >= 2: n! - 1 is prime if k<=n for smallest possible k. On 5 Jul 2017, at 19:03, Dan Asimov wrote:
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
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