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See also Math StackExchange: Is there a prime of the form $(9n)!+n!+1$?

Recently, user Peter from Math StackExchange asked for a prime of the form $(9n)!+n!+1$ (where $n$ is some natural number).

With the help of a computer he was able to find primes of the form $(kn)!+n!+1$ for $k=1,2,4,6,7,8$ and he was also able to find probable primes for $k=3,5$ (please see the StackExchange question linked above). Additionally, I was able to find that $n=237$ and $n=54$ give the smallest probable primes for $k=10$ and $k=11$ respectively.

However, for $k=9$, I checked every $n\le 2000$ using Mathematica with no prime found (!). Is there any reason for this or is it just „bad luck“?

Here is the Mathematica code I used:

SetSharedVariable[primes, checked]; primes = {}; checked = {};
Monitor[
 ParallelDo[
  If[! PrimeQ[n + 1],
   If[PrimeQ[(9 n)! + n! + 1], AppendTo[primes, n]]
  ];
  AppendTo[checked, n],
  {n, STARTHERE, STOPHERE}, Method -> "FinestGrained"
 ],
 {Sort[checked], primes}
]
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    $\begingroup$ For $n$ 1 less than a prime, the expression is 0 mod n+1. (This holds for all k.) You need to pick n large enough so that n! + 1 has no prime factors less than 9n. But this is necessary, and not sufficient. Gerhard "A Prime Observation for Compositeness" Paseman, 2019.12.09. $\endgroup$
    – Gerhard Paseman
    Commented Dec 10, 2019 at 0:52
  • $\begingroup$ The only idea, concerning mathematics, that I had in the past about this kind of problems is try to write a new equation using the composition of different arithmetic functions with the purpose to invoke inequalitites/conjectures that satisfy the arithmetic functions that I evoke. For example I cann't find a solution of $\varphi(m)=(9n)!+n!$, where $\varphi(m)$ is the Euler's totient function, for the first few positive integers $n\geq 1$ and $m\geq 1$. See the code for(m=1, 10000, for(n=1, 100, if(eulerphi(m)==(9*n)!+n!,print(m+1)))) in the web Sage Cell Server, choosing as Language GP. $\endgroup$
    – user142929
    Commented Dec 10, 2019 at 10:27
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    $\begingroup$ I check numbers of the form $(kn)!+n!+1$,It seems there's no special reason can't be primes.It's problem of probility. You bet,from $1$ to $n$, if you find 100 primes of the form $(8n)!+n!+1$, then from $1$ to $n$ you will find at least one prime of the form $(9n)!+n!+1$ $\endgroup$
    – Mike
    Commented Dec 10, 2019 at 10:33

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