Factorial primes are primes of the form $n!\pm1$. (In this application I'm interested specifically in $n!+1$ but any answer is likely to apply to both forms.) It seems hard to prove that there are infinitely many primes of this form, though Caldwell & Gallot were courageous enough to conjecture that there are infinitely many and to give a conjectured density ($e^\gamma\log x$ below $x$).

I'm looking at the opposite direction: how many composites are there of the form $n!\pm1$? It seems 'obvious' that the fraction of numbers of this form which are composite is 1, but I cannot even prove that there are infinitely many composites of this form.

Has this been proved? (Perhaps there's even a proof easy enough to relate here?) Or on the other hand, is it known to be open?

up vote 7 down vote accepted

Wilson's Theorem shows there are infinitely many composites. For if $p$ is prime, then $(p-1)!+1$ is divisible by $p$, and apart from the cases $p=2$ and $p=3$, the number $(p-1)!+1$ is greater than $p$.

There are related ways to produce a composite. For example, let $p$ be a prime of the form $4k+3$. Then one of $\left(\frac{p-1}{2}\right)!\pm 1$ is divisible by $p$.

  • Great. Do you think this can be expanded to show that the density of composites is positive? – Charles Apr 19 '13 at 18:31
  • The game with $4k+3$ can be extended, using the Dirichlet Theorem on primes in arithmetic progressions. But that still only produces a very thin set, undoubtedly very far from the truth. – André Nicolas Apr 19 '13 at 18:36
  • I don't understand how $p|\left(\frac{p-1}{2}\right)!$? Is it supposed to be $p|\left(\frac{p-1}{2}\right)!-1$? – Inceptio Apr 20 '13 at 9:37
  • 1
    Thanks for noticing the typo. A $\pm $ was left out, and for clarity I added "one of." – André Nicolas Apr 20 '13 at 9:53
  • @AndréNicolas: Your welcome. But still it doesn't prove for which primes $\left(\frac{p-1}{2}\right)!- 1$ is divisible by $p$. Its a more harder problem.(As discussed in the link I've posted) – Inceptio Apr 21 '13 at 8:34

Wilson's theorem:

$(p-1)! \equiv -1 \mod p$

Wilson's theorem can be used for finding infinite values for $n!+1$ being composite.

For the second set of numbers(i.e $n!-1$).Very famous conjecture was given by Louis J Mordell for the values of $p$ which satisfies $(\dfrac{p-1}{2})! \equiv 1 \mod p$ . About which you can read here.

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