Is $n! + 1$ often a prime?

$n! + 1$ is prime for $n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, \dots$, no other factorial primes are known as of May 2014. See here for more info on factorial primes.

Just looking at the heuristics of the problem:

If you pick a random integer $x$, it will be a prime number with a probability about $1 / \ln x$. Now the number $n! + 1$ is not a random integer. We know that $n! + 1$ is not divisible by any prime number $p ≤ n$. A random large integer is not divisible by any prime $p ≤ n$ with probability $(1-1/2)(1-1/3)(1-1/5)...$ which is about $1 / (2 \ln n)$. So the likelihood that $n! + 1$ is a prime is accordingly higher, about $2 \ln n / \ln (n!)$.

Using the Stirling formula, $\ln (n!)$ is about $n \ln n - n$ or $n(\ln n - 1)$. So $n!+1$ is prime with probability about $(2/n)/(1 - 1 / \ln n)$.

The factor $(1 - 1 / \ln n)$ is quite close to 1; the number of primes of the form $n! + 1$ with $n ≤ M$ is about $2 \ln M$. Very roughly agrees with the list of primes given earlier (I think it is a list of known primes, with many numbers in between not examined).

Such numbers are called factorial primes. There is only limited number of known such numbers.

The largest factorial primes are discovered only recently. From an announcement of an organization called PrimeGrid PRPNet:

On 30 August 2013, PrimeGrid’s PRPNet found the 2nd largest known Factorial prime: $$147855!-1$$ The prime is $700,177$ digits long. The discovery was made by Pietari Snow (Lumiukko) of Finland using an Intel(R) Core(TM) i7 CPU 940 @ 2.93GHz with 6 GB RAM running Linux. This computer took just a little over 69 hours and 37 minutes to complete the primality test.

PrimeGrid is a set of projects based on distributed computing, and devoted to finding primes satisfying various conditions.

Factorial primes-related recent events in PrimeGrid:

$147855!-1$ found: official announcement

$110059!+1$ found: official announcement

$103040!-1$ found: official announcement

$94550!-1$ found: official announcement

Other current PrimeGrid activities:

• 321 Prime Search: searching for mega primes of the form $3·2n±1$.
• Cullen-Woodall Search: searching for mega primes of forms $n·2n+1$ and $n·2n−1$.
• Extended Sierpinski Problem: helping solve the Extended Sierpinski Problem.
• Generalized Fermat Prime Search: searching for megaprimes of the form $b2n+1$.
• Prime Sierpinski Project: helping solve the Prime Sierpinski Problem.
• Proth Prime Search: searching for primes of the form $k·2n+1$.
• Seventeen or Bust: helping to solve the Sierpinski Problem.
• Sierpinski/Riesel Base 5: helping to solve the Sierpinski/Riesel Base 5 Problem.
• Sophie Germain Prime Search: searching for primes $p$ and $2p+1$.
• The Riesel problem: helping to solve the Riesel Problem.