If $P$ is prime then is $P+6$ prime?

Question

I would like to ask if it's proven that: "If $P\geq5$ is prime then $P+6$ or $P+12$ or $P+18$ or $P+30$ is prime"? If not is it likely to be true?

I assume you've already checked all the primes up to some value? How large? — Nate Eldredge, Feb 11 '19 at 01:00
https://math.stackexchange.com/questions/2311652/im-trying-to-find-the-longest-consecutive-set-of-composite-numbers?noredirect=1&lq=1 — John Wayland Bales, Feb 11 '19 at 01:07
For what it's worth the title question is entirely different from the question body. — fleablood, Feb 11 '19 at 03:17
Slightly extending the answer below : The difference between consecutive primes is not bounded. So, even if you add finite many expressions of the form $\ P+a\ $ with positive integer $\ a\ $, there still exists a counterexample , in fact infinite many. — Peter, Feb 11 '19 at 11:16

score 5 · Answer 1 · answered Feb 11 '19 at 01:05

Observe that $$2|(31! + 2), 3|(31! + 3), 4 | (31! + 4), \dots ,31|(31!+31).$$ Therefore, there are 30 consecutive composite numbers beginning at $31! + 2$. Now look at the biggest prime smaller than $31!+2$. It is followed by at least 30 non-primes, so we have a counterexample.

If $P$ is prime then is $P+6$ prime?

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