First prime before & after a factorial is a prime distance away?

[a nicol]

Excluding a gap of one, it seems that the first prime before and after a factorial is always a prime distance away.

For instance:
5! = 120, the next prime after is 127, this is 7 away, which is also prime.

This conjecture is unproven. So far it has been tested true up to 4000! for positive and 1000! for negative.

This does not apply to 2^n - the first prime after 2^n is not always a prime distance away. Why the difference?

Data:
a(n) = p-n!, where p is the smallest prime > n!+1.
https://oeis.org/A037153

[Gelly]

If it's not a prime distance away, it would have to be at least (n+1)^2.

Otherwise, n!+d will be divisible by the smallest prime factor of d, since the smallest prime factor of d will be smaller than n and, trivially, divide into n!.

The merit of a prime gap this size would be at least (n+1)^2 / ln(n!), or something like n/(ln(n)-1), which gets larger as n gets bigger. The largest prime gap merit ever found is 38.07, so for n = 150 and bigger, finding an example of a non-prime distance away from a factorial being the first prime would lead to record-breaking prime gap merits.

It's hyper-unlikely (and likely conjecturable) that an example of a non-prime gap away from a factorial being the first prime does not exist.

[Dr Sardonicus]

Let n > 2, and let Q be the largest prime less than n!. Let k be the least integer greater than 1 for which n! + k is prime.

Clearly k can be composed only of prime factors greater than n. If p is the least prime greater than n, the smallest composite k composed of primes greater than n is p^2.

So for k to be composite, the gap between Q and the next prime is at least p^2 > n^2. Now

log(n!) = n*log(n) apprixomately, so n = log(n!)/loglog(n!) approximately, so

p^2 = log^2(Q)/loglog^2(Q), approximately.

This would be (to say the least) unusually large for a gap between Q and the next prime, though not beyond the "infinitely often" size conjectured by Granville.

[a nicol]

There are no primes in the range [n!+2, n!+n] ?

[R. Gerbicz]

Originally Posted by Dr Sardonicus

p^2 = log^2(Q)/loglog^2(Q), approximately.

This would be (to say the least) unusually large for a gap between Q and the next prime, though not beyond the "infinitely often" size conjectured by Granville.

Very standard heuristic idea gives that we are expecting very few [maybe zero] composites for G=nextprime(n!)-n! [if we exclude the G<2 cases].

As you noted to find composite G we need G>n^2 so the [n!+1,n!+n^2] interval contains only composites. A random number near x is composite with 1-1/log(x) probability, so using log(n!)~n*log(n) we get composite G with at most (1-1/(n*log(n)))^(n^2) probability, that is approx. exp(-1/log(n)^2)
and the sum of these is convergent [elementary way: for fixed K sum of these in the exp(K)<n<exp(K+1) interval, giving that the sum of these is <sum(exp(K-K^2)) what is clearly a convergent serie.]

[a nicol]

Line graphs of A037153 in order and ascending numerical order:

https://oeis.org/A037153/a037153_1.png

[LaurV]

Originally Posted by a nicol

There are no primes in the range [n!+2, n!+n] ?

Nope. Any number of the form n!\(\pm\)x with x in [2..n] is divisible by x (because n! contains x as a factor). So, if none of the n!-1 and n!+1 is prime, then there are no primes in [n!-n, n!+n].

[SethTro]

These properties are used by the prime gap search.

Also true of primorials,

for P# if P# +-1 are not prime then there are no primes in [P#-P, P#+P]

For P#/2 if P# +- 2^n are not prime then there are no primes in [P#-2P, P#+2P]