Are all primes in a PAP-3? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:19:02Z http://mathoverflow.net/feeds/question/34197 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34197/are-all-primes-in-a-pap-3 Are all primes in a PAP-3? Charles 2010-08-02T03:06:52Z 2011-05-10T14:56:56Z <p>Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green &amp; Tao [2] famously extended this theorem to length $k$.)</p> <p>But taking this in a different direction, are <em>all</em> odd primes in a PAP-3? That is, for every prime $p>2$, is there a $k$ such that $p+k$ and $p+2k$ are prime?</p> <p>Unsurprisingly, the first 100,000 primes have this property; the largest value of $k$ needed is just 1584 (See [4], though the calculations don't go that far there). Heuristically, you'd expect a given prime to be in $$\int_2^\infty\frac{a\ dx}{\log(x\log x)}=+\infty$$ different PAP-3s, and there are no small prime obstructions, so the conclusion seems reasonable. On the other hand, it seems to involve Goldbach-like (or better, Sophie Germaine-like) additive patterns in the primes: in essence, we're looking for prime $q$, $2q-n$ for a fixed odd $n$, so I don't imagine this has been resolved.</p> <p>Basically, I'm just looking for more information on this problem. Surely it's been posed before, but does it have a common name and/or citation? Have any partial results been proved? Perhaps this is a consequence of a well-known conjecture?</p> <p>[1] A. G. van der Corput (1939). "Über Summen von Primzahlen und Primzahlquadraten", <em>Mathematische Annalen</em> <strong>116</strong>, pp. 1-50.</p> <p>[2] Ben Green and Terence Tao (2008). "The primes contain arbitrarily long arithmetic progressions", <em>Annals of Mathematics</em> <strong>167</strong>, pp. 481–547. <a href="http://arxiv.org/abs/math/0404188" rel="nofollow">http://arxiv.org/abs/math/0404188</a></p> <p>[3] Amarnath Murthy, <a href="http://oeis.org/A084704" rel="nofollow">http://oeis.org/A084704</a></p> <p>[4] Giovanni Teofilatto, <a href="http://oeis.org/A120627" rel="nofollow">http://oeis.org/A120627</a></p> http://mathoverflow.net/questions/34197/are-all-primes-in-a-pap-3/34298#34298 Answer by Ben Green for Are all primes in a PAP-3? Ben Green 2010-08-02T20:25:10Z 2010-08-02T20:25:10Z <p>This question is extremely close to this one</p> <p><a href="http://mathoverflow.net/questions/2214/covering-the-primes-by-3-term-aps" rel="nofollow">http://mathoverflow.net/questions/2214/covering-the-primes-by-3-term-aps</a></p> <p>though not exactly the same.</p> <p>For much the same reasons as described in the answer given there, the answer to your question is almost certainly yes, but a proof is beyond current technology, exactly as you suggest. I'm not aware that the problem has a specific name.</p> <p>To show that 3 belongs to a 3PAP is of course trivial: it belongs to 3,5,7 or 3,7,11. Showing that there are infinitely many such 3PAPs is, as you point out, a problem of the same level as difficulty as the Sophie Germain primes conjecture or the twin primes conjecture.</p> <p>For a general p, I find it extremely unlikely that you could show that there is a k > 0 such that p + k, p + 2k are both prime without showing that there are infinitely many. Proving this for even one value of p would be a huge advance.</p> <p>I think you could show that almost all primes p do have this property using the Hardy-Littlewood circle method.</p>