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

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 1:00
• The first counterexample is $463$. – Robert Israel Feb 11 '19 at 1:01
• I've looked at the first 10000 primes. – math Feb 11 '19 at 1:02
• @math: Then apparently you made a mistake...? – Nate Eldredge Feb 11 '19 at 1:04
• For what it's worth the title question is entirely different from the question body. – fleablood Feb 11 '19 at 3:17

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.