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I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.

Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ and $7$).

Is it known if $3$ is only value for $n$ or is it still an open problem ?

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    It's a well-known conjecture of Erdős for the $n!+1$ case, actually... – 2017-01-13
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    Take a look at this https://en.wikipedia.org/wiki/Factorial_prime – 2017-01-13
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    @kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime – 2017-01-13
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    Did you follow the link on wikipedia to here: https://oeis.org/A088054 ? – 2017-01-13
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    @Nate : Yes, and the only case i can see is for $n=3$ giving 5 and 7 – 2017-01-13
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    Did you read the line labeled "COMMENTS"? – 2017-01-13
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    @Nate : Yes, thanks, so it's conjectured but not proved, any other source ? – 2017-01-13
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    I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes. – 2017-01-13
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    In the question as asked, $n!\pm 1$ are twin primes. $6k\pm 1$ are twin primes iff $k\ne 6ab\pm a\pm b$. So if $n!\pm 1$ are never twin primes for $n>3$, then $\exists a,b\in \mathbb Z$ such that $\forall n>3\in \mathbb Z,\ \frac{n!}{6}=6ab\pm a\pm b$. I don't see a quick way to prove that, but I invite others to look into it. – 2018-11-21

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