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One of the steps in Euclid's proof of the infinity of primes is sometimes misinterpreted to be a way of generating new prime numbers. Specifically constructing the number P!+1 where P is a prime is sometimes though to guarantee a new prime by naive students.

But what is the ratio of prime factorials plus one that are actually prime (and does this ratio tend to a known limit)?

It is worth adding that, as far as I can tell (see this) only 4 such numbers are known to be primes so the ratio is likely to be very small!

matt_black
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  • A possibly naive response: Doesn't the factorial function just grow too fast for the ratio $\frac{|{p\leq n : p\text{ is a prime of the form }m!+1}|}{|{p \leq n : p \text{ is a prime}}|}$ to tend to anything other than $0$? The denominator grows at a rate of $n/\log n$ but doesn't the numerator grow extremely slowly? – James Apr 13 '16 at 17:51
  • @James Possibly. But I'm interested to see whether anything is known for sure and I'm not much of a mathematician. – matt_black Apr 13 '16 at 18:02
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    The OEIS sequences https://oeis.org/A088332 and https://oeis.org/A002981 are relevant. The sequence looks sparse enough that I'd guess the density is 0. The paper by Caldwell and Gallot at http://www.utm.edu/staff/caldwell/preprints/primorials.pdf gives a heuristic (Conjecture 2.3) that the number of primes $n! + 1$ with $n \le N$ is asymptotic to $e^\gamma \log N$ as $N \to \infty$. – Michael Lugo Apr 13 '16 at 18:03
  • Suppose that, for sufficiently large $p,$ $p!+1$ is never a prime? – daniel Apr 17 '16 at 05:13
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    It's not an arithmetic sequence and I'm pretty sure it's an open question whether there are infinitely many primes of this form. – daniel Apr 17 '16 at 05:18

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