TITLE: Is there a prime of the form $\ (9n)!+n!+1\ $?
QUESTION [17 upvotes]: See also MathOverflow: Why am I unable to find primes of the form $(9n)!+n!+1$?
In a project, I search primes of the form $$(kn)!+n!+1$$ with positive integers $\ k,n\ $. The smallest $\ k\ $ for which I still know no prime is $\ k=9\ $.
For $$k=1,2,3,4,5,6,7,8$$ the numbers $$n=1,3,605,185,850,7,11,120$$ are respective the smallest $n$ for which we get a prime, except possibly for $n = 605, 850$, in which case we just know we get a probable prime (the rest is proven to be prime according to FactorDB)

Is there a prime of the form $$(9n)!+n!+1$$ with positive integer $\ n\ $ ?

Chances should be good because such a number cannot have a prime factor less than or equal to $\ n\ $ , but upto $\ n=500\ $ , there is none.

REPLY [2 votes]: This may or may not help:
$$(kn)!+n!+1=\left(\prod_{m=2}^k\binom{mn}{n}\right){n!}^k+n!+1=n!\left(\left(\prod_{m=2}^k\binom{mn}{n}\right){n!}^{k-1}+1\right)+1$$ which is nearly recursive in $k$ Just chiming in .