Zhang's theorem and Polignac's conjecture

Question

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$.

Zhang's theorem has been significantly improved and, according to the Polymath8 project home page (http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes), the best unconditional bound to date is $246$.

Suppose the bound is lowered enough so as to prove the twin prime conjecture. If this happens, then one of the most famous unsolved problems in Mathematics will have been solved. But what will happen with Polignac's conjecture?

Polignac's conjecture states that for every positive integer $n$ there are infinitely many pairs of prime numbers whose difference is $2n$. So, since the twin prime conjecture is just a particular case of Polignac's conjecture, proving the former does not imply that the latter is true.

I know that probably now many mathematicians are trying to lower the bound enough so as to prove the twin prime conjecture. But, is any progress being made towards the solution of Polignac's conjecture? Can Zhang's discovery and his techniques be used in any way to make progress towards the solution of Polignac's conjecture, or should another groundbreaking discovery be made? Is now Polignac's conjecture closer to be proved or is it still "out of reach"?

Answer 1

Zhang proved that any admissible set of $k_0=3\,500\,000$ (or more) numbers contains some $a,b$ such that $n+a$ and $n+b$ are both prime infinitely often. There is an admissible set of this size with values between 0 and 70,000,000 thus the statement that there are infinitely many prime gaps at most 70 million.

The best value proved for $k_0$ so far is 50, which leads to the gap of 246 via the admissible tuple (0, 4, 6, 16, 30, 34, 36, 46, 48, 58, 60, 64, 70, 78, 84, 88, 90, 94, 100, 106, 108, 114, 118, 126, 130, 136, 144, 148, 150, 156, 160, 168, 174, 178, 184, 190, 196, 198, 204, 210, 214, 216, 220, 226, 228, 234, 238, 240, 244, 246).

But if you wanted you could choose a different tuple which showed, for example, that there are infinitely many prime gaps in a different range. For example, the admissible 50-tuple (0, 4, 10, 16, 22, 30, 34, 42, 46, 52, 60, 64, 70, 76, 84, 90, 94, 100, 106, 112, 126, 130, 136, 142, 150, 154, 160, 172, 184, 192, 202, 210, 214, 220, 226, 232, 240, 244, 252, 262, 270, 276, 280, 286, 294, 312, 316, 324, 330, 336) proves that there are infinitely many prime gaps of length between 4 and 336 (inclusive).*

So if the current methods were extended to prove the twin prime conjecture it would automatically prove Polignac's conjecture. Now that might be too much to expect -- the Polymath project has already changed its methodology in significant ways in the course of its several months of operation. But it does serve to show that Polignac's conjecture is not far from the twin prime conjecture.

A reasonable question, then, is "can Zhang's method be so extended?". At the moment the answer seems to be "no": even on the assumption of the generalized Elliott-Halberstam conjecture, the best that has been achieved is $k_0=3$ which means (via the 3-tuple (0, 2, 6)) that at least one of twin primes, cousin primes, and sexy primes have infinitely many members. But even with that high-powered assumption we can't narrow it down further.

* Similarly I can show that there are infinitely many prime gaps between 6 and 378, between 8 and 502, between 10 and 616, between 12 and 678, and so forth. On GEH the best you can do is $g$ to $2g$ if $3|g$ or $g$ to $2g+2$ otherwise.