Let $P$ denote the set of primes in $\mathbb{N}$. For $k\in \mathbb{N}, k\geq 2$ set $$M_k = \big\{p\in P: \{kp-1, kp+1\}\cap P \neq \emptyset\big\}.$$ Is there $k\in \mathbb{N}, k\geq 2$ such that $M_k$ is infinite?
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asked Jul 21, 2017 at 4:15
Dominic van der Zypen
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$\begingroup$ I would assume one can prove that $\bigcup\{M_k:k\in\mathbb{N}, k\ge 2\}$ is infinite. $\endgroup$Dominic van der Zypen– Dominic van der Zypen2017-07-21 04:16:07 +00:00Commented Jul 21, 2017 at 4:16
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1$\begingroup$ Yes, in fact for each prime $p$, Dirichlet's theorem says there are infinitely many primes of the form $kp-1$ and infinitely many of the form $kp+1$, so $\bigcup_{k \ge 2} M_k$ consists of all the primes. $\endgroup$Robert Israel– Robert Israel2017-07-21 05:26:49 +00:00Commented Jul 21, 2017 at 5:26
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$\begingroup$ Your question is in some sense dual to asking whether for some prime $ p $ , the set $ W_{p} : =\{k\in\mathbb{N}\colon\{kp-1,kp+1\}\cap P\neq\emptyset\} $, is infinite. For $ p =3$, a positive answer follows from the assumption of the twin prime conjecture. $\endgroup$Sylvain JULIEN– Sylvain JULIEN2017-07-21 21:19:32 +00:00Commented Jul 21, 2017 at 21:19
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1 Answer
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You're asking whether, for some $k \ge 2$, there are infinitely many primes $p$ such that either $kp-1$ or $kp+1$ is prime. That would mean there are infinitely many primes such that $kp-1$ is prime or infinitely many primes such that $kp+1$ is prime. Of course, $k$ had better be even. If $k$ is even, Dickson's conjecture would say that in both cases the answer is yes.
But Dickson's is still a conjecture, and there are no $k$ for which the answer has been proven to be yes.
