This is a question related to my recent question Conjecture: “For every prime $k$ there will be at least one prime of the form $n!\pm k$” true?
Using PARI/GP I searched for the number of primes of the form $n!+k$ possible for each prime $k\le 2300$. I observed that for some prime $k$, all numbers less than $24$ were found except $2$.
Note that when $n \ge k$, $n! \pm k$ cannot be prime as $k$ will be a factor of $n! \pm k$. This means that there are a finite number of primes of the form $n! \pm k$ for each $k$.
User Peter and I, searched for some prime $k$ for which $n!+k$ has two primes upto $k\le 10^5$ and found none in the range. Here is the code we used in PARI/GP if anyone wants to extend the search:
forprime(k=2,10^7,s=0;for(n=1,k,if(s<3,if(ispseudoprime(n!+k)==1,s=s+1)));if(s==2,print(k)))
Conjecture:
For any prime $k$ there cannot be exactly two primes of the form $n!+k$.
Can prove/disprove this? Any ideas, heuristics, guesses are welcome.
Update: Upto $k \le 10^6$, $k=714377$ is the only candidate remaining for which we have only found one prime of the form $n!+k$ till $n\lt 3000$. If a second prime of the form $n!+714377$ is found for the range $3000 \lt n \lt 714377$, then the conjecture is disproved!