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Prove that there are infinitely many primes $P$, such that $3P+2$ and $\frac{3P+1}{2}$ are also primes. Something possible to do?

Dietrich Burde
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zucchini
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    You can show there are Infinitely many primes of the form $3k+2$ when $k$ is any integer. However the statement you want to prove is very likely unknown (see e.g. https://math.stackexchange.com/questions/911690/are-there-infinite-many-primes-p-such-that-2p-1-is-also-prime). – Winther Oct 04 '19 at 11:56
  • The first few such primes are $$3,7,19,59,127,167,239,439,479,607,859,967,1039,1259,1427,1559,1567,1699,1879,1979$$ but this sequence is not in OEIS. It's the intersection of https://oeis.org/A023208 and https://oeis.org/A158709. – lhf Oct 04 '19 at 12:49
  • How did you get those sequence? – zucchini Oct 04 '19 at 12:59
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    This has been an unsolved problem for nearly $200$ years. So you should not expect a solution in the next few hours. – Nilotpal Sinha Oct 04 '19 at 13:49

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(3P+1)/2 would be a special Sophie Germain prime, which is a prime p and 2p+1 is also a prime. It is not known if there exist infinitely many Sophie Germain primes. Therefore it is not known if there exist infintely many P.

HellsKitchen
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