Are there infinitely many pairs of primes of the form $p,2p-1$?
What about $p,2p+1$?
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1The latter type are called Sophi-Germain primes. See the Wikipedia page. It is not known if there are infinitely many such pairs. – Barry Smith Aug 24 '15 at 00:16
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https://en.wikipedia.org/wiki/Sophie_Germain_prime – mich95 Aug 24 '15 at 00:16
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https://en.wikipedia.org/wiki/Dickson%27s_conjecture – Robert Israel Aug 24 '15 at 00:21
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See OEIS A$005382$. – Lucian Aug 24 '15 at 00:22
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is 2p-1 also an accepted value? – Moti Aug 24 '15 at 00:32
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Accepted in terms of Dickson's conjecture? Yes. – Robert Israel Aug 24 '15 at 00:39
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However, for any $\epsilon > 0$, there are infinitely many pairs of primes $p, q$ such that $|\frac{p}{q}-2| < \epsilon $.
This is a very special case of the result proved in Hobby, D., D. M. Silbeger, Quotients of primes, Amer. Math. Monthly, Vol. 100, 1993, No. 1, 50–52, that the ratio $\frac{p}{q}$ taken over all pairs of primes is dense in the positive reals.
marty cohen
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