The "bounded gaps between primes" Polymath project - a retrospective

Abstract

For any $m\ge 1$, let $H_m$ denote the quantity $H_m:={\mathrm{liminf}}_{n\to \infty }(p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\mathrm{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is equal to two. In a recent breakthrough paper of Zhang, a finite upper bound was obtained for the first time on $H_1$; more specifically, Zhang showed that $H_1\le 70000000$. Almost immediately after the appearance of Zhang's paper, improvements to the upper bound on $H_1$ were made. In order to pool together these various efforts, a \emph{Polymath project} was formed to collectively examine all aspects of Zhang's arguments, and to optimize the resulting bound on $H_1$ as much as possible. After several months of intensive activity, conducted online in blogs and wiki pages, the upper bound was improved to $H_1\le 4680$. As these results were being written up, a further breakthrough was introduced by Maynard, who found a simpler sieve-theoretic argument that gave the improved bound $H_1\le 600$, and also showed for the first time that $H_m$ was finite for all $m$. The polymath project, now with Maynard's assistance, then began work on improving these bounds, eventually obtaining the bound $H_1\le 246$, as well as a number of additional results, both conditional and unconditional, on $H_m$. In this article, we collect the perspectives of several of the participants to these Polymath projects, in order to form a case study of online collaborative mathematical activity, and to speculate on the suitability of such an online model for other mathematical research projects.

19 pages, submitted, Newsletter of the EMS

References in corpus (11)

• Bounded gaps between primes with a given primitive root
• Bounded Gaps Between Primes in Chebotarev Sets
• On the distribution of gaps between consecutive primes
• Bounded gaps between primes of the special form
• Consecutive primes in tuples
• Bounded length intervals containing two primes and an almost-prime
• A note on the theorem of Maynard and Tao
• Arithmetic functions at consecutive shifted primes
• On the hot spots conjecture for acute triangles
• A note on bounded gaps between primes
• Bounded prime gaps in short intervals

Cited by in corpus (3)

• A Possible Solution for Hilbert's Unsolved 8th Problem: Twin Prime Conjecture
• A "Vertical" Generalization of the binary Goldbach's Conjecture as applied on primes with prime indexes of any order i (i-primes)
• Expanding total sieve and patterns in primes