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], 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.

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/classic/A084704http://oeis.org/A084704

[4] Giovanni Teofilatto, http://oeis.org/classic/A120627http://oeis.org/A120627

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# Are all primes in a PAP-3?

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], 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.

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/classic/A084704
[4] Giovanni Teofilatto, http://oeis.org/classic/A120627