nt.number theory - Linear equation with primes

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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