Let $p_n$ be the $n$-th prime number, as usual: $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.

For $k=1,2,3,\ldots$, define $$ g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n). $$ Thus the twin prime conjecture asserts $g_1 = 2$.

Zhang's theorem (= weak twin prime conjecture) asserts $g_1 < \infty$.

The prime $m$-tuple conjecture asserts $g_2 = 6$ (infinitely many prime triplets), $g_3 = 8$ (infinitely many prime quadruplets), "etcetera" (with $m=k+1$).

Can Zhang's method be adapted or extended to prove $g_k < \infty$ for any (all) $k>1$?

*Added a day later*: Thanks for all the informative comments and answers!
To summarize and update (I hope I'm attributing correctly):

- [Eric Naslund] The question was already raised in the Goldston-Pintz-Yıldırım paper. See Question 3 on page 3:

Assuming the Elliott-Halberstam conjecture, can it be proved that there are three or more primes in admissible $k$-tuples with large enough $k$? Even under the strongest assumptions, our method fails to prove anything about more than two primes in a given tuple.

[several respondents] As things stand now, it does not seem that Zhang's technique or any other known method can prove finiteness of $g_k$ for $k > 1$. The main novelty of Zhang's proof is a cleverly weakened estimate a la Elliott-Halberstam, which is well short of "the strongest assumptions" mentioned by G-P-Y.

[GH] For $k>1$, the state of the art remains for now as it was pre-Zhang, giving nontrivial bounds not on $g_k$ but on $$ \Delta_k := \liminf_{n \rightarrow \infty} \frac{p_{n+k} - p_n}{\log n}. $$ The Prime Number Theorem (or even Čebyšev's technique) trivially yields $\Delta_k \leq k$ for all $k$; anything less than that is nontrivial. Bombieri and Davenport obtained $\Delta_k \leq k - \frac12$; the current record is $\Delta_k \leq e^{-\gamma} (k^{1/2}-1)^2$. This is positive for $k>1$ (though quite small for $k=2$ and $k=3$, at about $0.1$ and $0.3$), and for $k \rightarrow \infty$ is asymptotic to $e^{-\gamma} k$ with $e^{-\gamma} \approx 0.56146$.

[Nick Gill, David Roberts] Some other relevant links:

Terry Tao's June 3 exposition of Zhang's result and the work leading up to it;

The "Secret Blogging Seminar" entry and thread that has already brought the bound on $g_1$ from Zhang's original $7 \cdot 10^7$ down to below $5 \cdot 10^6$;A PolyMath page that's keeping track of these improvements with links to the original arguments, supporting computer code, etc.;

A Polymath proposal that includes the sub-project of achieving further such improvements.

- [Johan Andersson] A warning: phrases such as
"large prime tuples in a given [length] interval"
(from the Polymath proposal) refer not to configurations
that we can prove arise in the primes but to
*admissible*configurations, i.e. patterns of integers that*could*all be prime (and*should*all be prime infinitely often, according to the generalized prime $m$-tuple [a.k.a. weak Hardy-Littlewood] conjecture, which we don't seem to be close to proving yet). Despite appearances, such phrasings do not bear on a proof of $g_k < \infty$ for $k>1$, at least not yet.

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