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It is a well-known conjecture that there exist infinitely many primes of the form $n!+1$.

This problem is somewhat similar to the famous Landau's problem about the infinitude of the primes of the form $n^2+1$. Iwaniec has shown that $n^2+1$ is a product of at most two primes for infinitely many $n$. Is there a similar result regarding $n!+1$?

For example, is there any partial progress towards this problem which places some bound on $\omega(n!+1)$ for infinitely many $n$?

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    $\begingroup$ The sequence $n! + 1$ grows far faster than $n^2 + 1$, and the modern methods of sieve theory cannot give results for such thin sequences as far as I know. For example, we can't prove whether the sequence $2^n - 1$ (a far fatter sequence) is square-free infinitely often, let alone a product of a bounded number of primes. $\endgroup$
    – Stanley Yao Xiao
    Commented Jul 4, 2023 at 23:29

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