After Zhang established the result that there are infinitely many pairs of primes separated by at most 70000000, there has been a massively collaborative effort to reduce this bound. The efforts are concentrated on developing methods to strengthen each of three variables in the original paper:

- The pair of real parameters
**(ω, δ)**, which were equal in the original paper, but later separated so that they can be optimised independently. Current efforts are to increase the value of ω in the hope of obtaining a smaller value of: - The integer
: This is a positive integer such that for each admissible set*k*0*S*of*k*0 integers, there are infinitely many translates*S*+*n*containing at least two primes. An*admissible*set is one where, for each prime*p*, there exists a residue class disjoint from*S*. At the moment, there is an established upper bound of 26024 and a conditional result by Terry Tao*et al*claiming that . Reducing the value of*k*0 leads to a smaller value of: - The integer
, which is the diameter of the smallest admissible set of size*H**k*0. For each*k*0, it is possible to find upper bounds (using sieves to construct admissible sets) and lower bounds (obviously,*k*0 is a trivial lower bound, but certain inequalities such as Montgomery—Vaughan provide much tighter bounds). More details are here.

For the last few hours, the progress has rather resembled a game of tennis between Andrew Sutherland and *xfxie*, with the world record for *H* being repeatedly tossed between them. At the moment, Sutherland is winning, having reduced *H* down to 285278. You can get a slice of the action by viewing this comments thread, and perhaps chiming in with a smaller admissible set*.

As you can see, Sutherland has introduced a new optimisation together with a rather colourful piece of new terminology to describe it! This technique can sometimes lead to incremental (or should that be excremental?) reductions in the upper bound for *H*.

* Entries are to be given as a list with the filename ** admissXble_YYYYY_ZZZZZZ.txt**, where

*X*is a vowel used to specify the author (

*i*for

*xfx*and

**i**e*a*for Sutherl

**a**nd; you can choose any other vowel in your surname),

*YYYYY*is the size of the set (preferably one of the established or conditionally proved values of

*k*0), and

*ZZZZZZ*is the diameter (an upper bound for

*H*, assuming your value of

*k0*is correct). Examples are admissible_26024_286216.txt and admissable_10719_108514.txt.

We’re still waiting for confirmation that the value 10719 is valid; if so, the upper bound on *H* will drop from 285278 to 108514 — a significant improvement, but still considerably far from the desired target of 2. Here’s the timeline at the moment:

For continually-updated data, consult the polymath page.

It’s great how a bit of competition can lead to such good results. We are 2 orders of magnitude closer to twin prime conjecture!

In related news, a new Olympic sport has been created.

248970 and still counting (backwards) …

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down to 6,966 confirmed via Maple by Terence Tao (k=0 via Hannes & H = 6,966 via Engelsma) – http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236517