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Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$?

The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ is not.

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    $\begingroup$ Any particular reason for the downvotes? $\endgroup$
    – Lucia
    Commented Nov 18, 2015 at 16:40
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    $\begingroup$ @Lucia: I guess you like my edit. $\endgroup$
    – Douglas Zare
    Commented Nov 18, 2015 at 16:48
  • $\begingroup$ @DouglasZare: Ok, that makes sense! And yes, I like your edit! $\endgroup$
    – Lucia
    Commented Nov 18, 2015 at 17:21

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A paper by Kisilevsky and Rubinstein (available here: http://arxiv.org/abs/1112.4945) shows that any set of primes determined by ``Chebotarev conditions'' can't be too regularly distributed within the primes. The set of primes in any union of residue classes (e.g. $2 \pmod{3})$ is a Chebotarev set of primes, so can't be too regularly distributed within the primes as a whole. Meanwhile, the set of primes of odd index are very regularly distributed within the primes, so they can't be a Chebotarev set.

See Corollary 1 in the linked paper.

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It was known since Littlewood, "Distribution des Nombres Premiers", C. R. Acad. Sci. Paris 158 (1914), 1869-1872, that there are infinitely many $x$ for which $\pi(x,3,1)>\pi(x,3,2)$ as well as infinitely many $x$ for which $\pi(x,3,2)>\pi(x,3,1)$. This implies that there can't be only finitely many primes $p_{2n-1}$ that are $2\pmod{3}$. For a more detailed account of more powerful and accurate results check out the Rubinstein, Sarnak paper on the Chebyshev bias.

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