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Is there an integer $n$ with an infinite number of representations of the form $n=2q-p$, where $p$ and $q$ are both primes?

Given a positive integer $k>1$, I would like to know for which (if any) integers $n$ the linear equation $q-kp=n$ admits an infinite number of solutions, where $p$ and $q$ are primes. (I'm not including $k=1$ because it reduces to well know open problems, $k=1$ and $n=2$ would be the twin primes conjecture)

The density of the prime numbers implies that at least there are integers $n$ with an arbitrarily large number of representations.

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Do you have any reason to expect that this is easier than the twin prime conjecture? – Qiaochu Yuan Dec 31 '09 at 3:49
$k=2$, $n=1$ is another well-known open problem (Sophie Germain primes) and so are all other cases except the ones when trivial considerations modulo some number garantee that there may be only finitely many solutions. I will be genuinely surprised if a technique is invented that will allow to solve some but not all nontrivial questions in this series. – fedja Dec 31 '09 at 3:49
Come on, guys, varying $k$ is just the Dirichlet's theorem about primes in arithmetic progressions! Of course, neither I, nor Manuel meant that. – fedja Dec 31 '09 at 4:07
Ah, very true, I'll head back and delete those comments. If $k$ is fixed, then this is a notoriously difficult problem. – Ben Weiss Dec 31 '09 at 4:17
Well, there is the work of Tao and Green about linear equations in primes, but it does not contain these cases… – Gjergji Zaimi Dec 31 '09 at 4:36
up vote 13 down vote accepted

Assuming the Hardy-Littlewood prime tuples conjecture, any n which is coprime to k will have infinitely many representations of the form q-kp.

Assuming the Elliot-Halberstam conjecture, the work of Goldston-Pintz-Yildirim on prime gaps (which, among other things, shows infinitely many solutions to 0 < q-p <= 16) should also imply the existence of some n with infinitely many representations of the form q-kp for each k (and with a reasonable upper bound on n). [UPDATE, much later: Now that I understand the Goldston-Pintz-Yildirim argument much better, I retract this claim; the GPY argument (combined with the more recent methods of Zhang) would be able to produce infinitely many $m$ such that at least two of $m + h_i$ and $km + h'_i$ are prime for some suitably admissible $h_i$ and $h'_i$, but this does not quite show that $q-kp$ is bounded for infinitely many $p,q$, because the two primes produced by GPY could both be of the form $m+h_i$ or both of the form $km+h'_i$. So it is actually quite an interesting open question as to whether some modification of the GPY+Zhang methods could give a result of this form.]

Unconditionally, I doubt one can say very much with current technology. For any N, one can use the circle method to show that almost all numbers of size o(N) coprime to k have roughly the expected number of representations of the form q-kp with q,p = O(N). However we cannot yet rule out the (very unlikely) possibility that as N increases, the small set of exceptional integers with no representations covers all the small numbers, and eventually grows to encompass all numbers as N goes to infinity.

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