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For some numbers $n!+1$ is prime, but all such numbers are not prime. For example, $5!+1 = 11\times 11$. The question is this:

Are there infinitely many primes of the form $n!+1$?

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At this time it is unknown. These are called (together with cases $n!-1$) factorial primes.

Chris Caldwell maintains a page of Factorial Prime Records, with links to much else.

Caldwell and Yves Gallot have a heuristic argument that there should be infinitely many factorial primes of each form (see their preprint here), with the estimate that the chance of $n! \pm 1$ of either form being prime is:

$$ \frac{e^\gamma \log n}{n(\log n - 1)} $$

Integrating this leads to their conjecture that "The expected number of factorial primes of each of the forms $n! \pm 1$ with $n \le N$ are both asymptotic to $e^\gamma \log N$."

Clearly this tends to infinity as $N \to \infty$.

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This is still an open problem, which is to determine whether there exists infinitely many factorial primes (primes of the form $n!\pm1$), according to this page.

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Please see answers to another related question here (Is n!+1 often a prime?).

There you can learn:

  • what factorial primes are, and how many are known to us at the present moment
  • some probability heuristics about existence and frequency of factorial primes
  • some collaborative computation efforts to find more factorial primes

Hope this would help.

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