Two weeks ago, Yitang Zhang announced his result establishing that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Since then there has been a flurry of activity in reducing this bound, with the current record being 4,802,222 (but likely to improve at least by a little bit in the near future).
It seems that this naturally suggests a Polymath project with two interrelated goals:
- Further improving the numerical upper bound on gaps between primes; and
- Understanding and clarifying Zhang’s argument (and other related literature, e.g. the work of Bombieri, Fouvry, Friedlander, and Iwaniec on variants of the Elliott-Halberstam conjecture).
Part 1 of this project splits off into somewhat independent sub-projects:
- Finding narrow
primeadmissible tuples of a given cardinality (or, dually, finding large primeadmissible tuples in a given interval). This part of the project would be relatively elementary in nature, relying on combinatorics, elementary number theory, computer search, and perhaps some clever algorithm design. (Scott Morrison has already been hosting a de facto project of this form at this page, and is happy to continue doing so).
- Solving a calculus of variations problem associated with the Goldston-Yildirim-Pintz argument (discussed at this blog post, or in this older survey of Soundararajan) [in particular, this could lead to an improvement of a certain key parameter , currently at 341,640, even without any improvement in the parameter mentioned in part 3. below.]
- Delving through the “hard” part of Zhang’s paper in order to improve the value of a certain key parameter (which Zhang sets at 1/1168, but is likely to be enlargeable).
Part 2 of this project could be run as an online reading seminar, similar to the online reading seminar of the Furstenberg-Katznelson paper that was part of the Polymath1 project. It would likely focus on the second half of Zhang’s paper and would fit well with part 1.3. I could run this on my blog, and this existing blog post of mine could be used for part 1.2.
As with other polymath projects, it is conceivable that enough results are obtained to justify publishing one or more articles (which, traditionally, we would publish under the D.H.J. Polymath pseudonym). But it is perhaps premature to discuss this possibility at this early stage of the process.
Anyway, I would be interested to gauge the level of interest and likely participation in these projects, together with any suggestions for improving the proposal or other feedback.