# Primes $p$ for which $2p-1$ is prime

It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?

Seemingly it's also an open problem (see here and the linked question). I am aware that it is a special case of more general open problems.

But I'd like a concrete reference asserting that this specific problem is indeed open.

Edit: As I said, I'm not interested in more general conjectures. It's suitable for MO, because no one has given me a reference for this exact problem.

• It's a special case of the Bateman-Horn conjecture. ams.org/journals/mcom/1962-16-079/S0025-5718-1962-0148632-7/… – Felipe Voloch Oct 10 '15 at 12:11
• Or Schinzel's hypothesis H. I think the problem is not lack of references but too many. Not really suitable for MO. – Felipe Voloch Oct 10 '15 at 12:13
• I think that the paper linked to by Carlo Beenakker pretty directly answers your question. It states "In this paper we focus our attention on Cunningham chains of length 2 of the second kind. As of January 2013 the largest known prime combination has 44 652 decimal digits." These are exactly the types of numbers you are looking for and the formulation is chose to convey it is not known there are infinitely many – user9072 Oct 10 '15 at 16:22
• The original version said "Seemingly it's also an open problem [...]. But I'd like a reference." Then it's not completely clear to me what type of reference you are looking for. Your comments clarified that you are looking for a reference for the fact that it is an open problem. But you could have just as well been interested in a discussion of methods or progress towards this problem. In the latter case the more general references would be helpful. Indeed, requests of the form: "I want more information on the problem." are in my experience more common than "I want it asserted it is open." – user9072 Oct 10 '15 at 19:47
• @user236182: What I tried to point out is this: if you can solve one $k=2$ case of Dickson't conjecture, then you can probably solve all $k=2$ cases such as the twin prime conjecture or the case of Sophie Germain primes. This is clear to any expert who worked on these kinds of problems. Mathematics is not a formal game, conjectures and opinions are not born in a vacuum. It is a fact that your question is open: I and other users of this site are the reference if you wish. I could go ahead and publish a paper stating this, would it be better? – GH from MO Oct 10 '15 at 22:10

I agree with Felipe Voloch that this question is not quite suitable for MO. At any rate, it is well-known (folklore) among number theorists that solving any single $k=2$ case of Dickson's conjecture would be a major breakthrough. The best approximations we know are in the works of Green-Tao, Zhang-Maynard-Tao-Polymath (and studying these works makes one understand why any single $k=2$ case is hard).
This is the "Cunningham chain of the second kind of length 2", here is the OEIS reference and here is the Wikipedia entry; record primes of this type are listed here, the currently largest known (52138 digits) prime $p$ such that $2p-1$ is also prime is $p=129431439657 \cdot 2^{170171} + 1$. It is a (still open) conjecture that the number of primes of this type less than $N$ grows with bound as $1.32\, N/\log^2 N$, see Sieving for large twin primes and Cunningham chains of length 2 of the second kind (2012).