Interval of n non-primes.

Question

I saw this question showing how you can generate a sequence of n non-prime numbers in a row: Intervals that are free of primes

It would be of the form: $$(n+2)! + 2, (n+2)! + 3, \ldots, (n+2)! + (n+2)$$

I was wondering about a similar interval but with the condition that it must be surrounded by primes. The term before is prime and the term after is also prime. This can only be true when n is odd as if it was even either the beginning or last term would have to be even and therefore it would not be surrounded by primes.

Eg. the condition would be met when:
n=1 -> 3,4,5
n=3 -> 13,14,15,16,17
n=5 -> 23,24,25,26,27,28,29

I was wondering if it is possible to prove that such an interval exists for all odd numbers n.

One minor point is that your interval of $(n+2)! + 2, (n+2)! + 3, \ldots, (n+2)! + (n+2)$ has $n+1$ non-prime numbers in a row, instead of the $n$ you want, so you should change the $n+2$ to $n+1$. — John Omielan, Dec 02 '23 at 02:42
What you're basically asking about is if, for every even number $m$, there's a prime gap of size $m$. Note that Polignac's conjecture proposes something considerably stronger, namely that each such even $m$ prime gap size occurs infinitely often. — John Omielan, Dec 02 '23 at 02:48
Does this answer your question? Can every even integer be expressed as the difference of two primes? - using a site search, the top-listed result of Are there prime gaps of every size? had, in its answer, a link to the proposed duplicate question. — John Omielan, Dec 02 '23 at 02:51
@JohnOmielan. Yep, I think so thanks for your help. So that conjecture remains unsolved correct? It seems to me to be an extension of the twin prime conjecture. So I think I'm a little out of my depth haha. — Tom, Dec 02 '23 at 02:53
You're welcome. As far as I know, it's still an open question. I've not heard or read about it being solved. Also, the dup. question's answer has a link to Prime Conjectures and Open Questions, where its third conjecture is "Every even number is the difference of two primes". Note below this it states "Chen's work mentioned in the discussion of the Goldbach conjecture also showed that every even number is the difference between a prime and a $P_2$", where $P_2$ means a semi-prime. This, I suspect, is the closest we are to solving it. — John Omielan, Dec 02 '23 at 02:59

Interval of n non-primes.

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