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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?

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    $\begingroup$ I assume you've already checked all the primes up to some value? How large? $\endgroup$ – Nate Eldredge Feb 11 at 1:00
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    $\begingroup$ The first counterexample is $463$. $\endgroup$ – Robert Israel Feb 11 at 1:01
  • $\begingroup$ I've looked at the first 10000 primes. $\endgroup$ – math Feb 11 at 1:02
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    $\begingroup$ @math: Then apparently you made a mistake...? $\endgroup$ – Nate Eldredge Feb 11 at 1:04
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    $\begingroup$ For what it's worth the title question is entirely different from the question body. $\endgroup$ – fleablood Feb 11 at 3:17
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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.

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