Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.)
But taking this in a different direction, are all 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?
Unsurprisingly, the first 100,000 primes have this property; the largest value of $k$ needed is just 1584 (see [4] and also [5] where this is greatly extended). 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.
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?
[1] A. G. van der Corput (1939). "Über Summen von Primzahlen und Primzahlquadraten", Mathematische Annalen 116, pp. 1-50.
[2] Ben Green and Terence Tao (2008). "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167, pp. 481–547. http://arxiv.org/abs/math/0404188
[3] Amarnath Murthy, http://oeis.org/A084704
[4] Giovanni Teofilatto, http://oeis.org/A120627
[5] Charles R Greathouse IV, https://oeis.org/A190423
Best Answer
This question is extremely close to this one
Covering the primes by 3-term APs ?
though not exactly the same.
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.
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.
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.
I think you could show that almost all primes p do have this property using the Hardy-Littlewood circle method.