Notes and links for the prime gaps backstory

31st May 2021 This page is currently under development. Please be patient. For further information please contact . Thanks to Keith Mathews for some good edits.

The human side of the prime gaps discoveries has many components - group and individual creativity, generous leadership, courage in the face of error, determination to solve problems over decades, rivalry and disappointment, deserved and undeserved fame, excellent and poor communication, generosity in giving credit and obstinacy when it comes to attribution. This web page gives some information and further links. I have started collecting these: for example articles in magazines, Journals and newsletters including some by main Polymath contributors, a documentary film, a book, and of a link to the Polymath8's home page, etc. Suggestions for additional links are most welcome. Finally, this web based information should not in anyway take the place of a coherent write-up, preparation for which would involve extensive interviewing as well as investigating primary and secondary sources. This is intended to complement the technical work "Bounded gaps between primes: the epic breakthroughs of the early 21st century".

Should a particular link become stale please contact the author of the article and also

Goldston, Motohashi, Pintz and Yildirim (GMPY)

(1) An article in Science Magazine by Dana MacKenzie announcing Goldston and Yildirim's initial result and recording the reactions of leading number theorists: Prime Proof helps mathematicians mind the gaps, Science 300 (2003) p32.

(2) An article in Science Magazine by Dana MacKenzie reporting on the mistake in the proof (1) discovered by Granville and Soundararajan: "Prime-number proof's leap falls short, Science 300 (2003) p1066.

(3) An article in Science Magazine by Barry Cipra announcing the Green Tao theorem on the existence of arithmetic progressions of primes of arbitrary finite length using a correct part of the work of Goldston and Yildirim (1): Proof promises progress in prime progressions, Science 304 (2004) p1095 .

(4) An article in Science Magazine by Barry Cipra announcing the complete proof of the theorem of Goldston, Pintz and Yildirim that for every fraction of the average distance between primes up to any given x there is at least one pair of primes less than that distance apart: Third time proves charm for prime-gap theorem, Science 308 (2005) p1238. This article gives some indication of the role of others, including Granville, Soundararajan, Motohashi and how the proof evolved.

(5) An expository article by Yoichi Motohashi The twin prime conjecture.

(6) Part of an article (on line) by Dan Goldston giving background historical information on how the GPY proof was found, giving information on the trial and error approach used in joint work with Yildirim, emphasizing the role played by Pintz in discovering the "l-trick" (replacing a k-tuple by a (k+l)-tuple), and the importance of having a symbolic computation assistant (Mathematica in his case). It also in Section 7, "A curious history", he shows how their ideas could be seen in a reduced form in earlier writings of Selberg and Heath-Brown, and the importance of the contributions of Granville and Soundararajan: Are there infinitely many twin primes?

(7) GMPY Backstory timeline:

Early 1990's: Goldston worked on short divisor sums giving lower bounds of the right order for second moments for primes.

1999: Yildirim joined Goldston in evaluating third moments.

2003-2004: Goldston and Yildirim applied their methods to small gaps between primes.

March 2003: Goldston announced their results at Oberwolfach.

April 2003: Granville and Soundararajan discovered an error in a lemma which was part of the derivation of Goldston and Yildirim.

May 2004: Granville pointed Green and Tao to a correct lemma in the (at that time unpublished) work of Goldston and Yildirim which enabled Green and Tao to complete the proof that arithmetic progressions of primes of any specified finite length (and thus an infinite number) exist.

Late 2004 and early 2005: Pintz joined Golston and Yildirim and provided one more idea (replacing k by k+l) to complete their proof.

February 2005: on receiving a draft of the complete proof, Motohashi (who was at the time doing work of great significance on matters related to bounded gaps between primes - see Zhang below) found a way of simplifying the argument for the main result.

Motohashi's work led to a presentation by Goldston at the May 2005 CUNY number theory conference and later publication (2006) in the Proceedings of the Japan Academy with authors Goldston, Motohashi, Pintz and Yildirim.

The full GPY paper, which was based on double tuples, and in particular enabled results on prime gaps assuming the Elliot-Halberstam conjecture, was published in the Annals of Mathematics in 2009.


(1) A movie which has an extended interview with Yitang Zhang and other interviews with some of the main contributors to the developments. It was produced and directed by George Csicery: Counting from infinity: Yitang Zhang and the twin primes conjecture, Zala films, Oakland, CA, 2015.

(2) A short article dealing with in particular the reviewing process for Zhang's Annals of Mathematics paper by John Friedlander: Prime numbers: a much needed gap is finally found, Notices Amer. Math. Soc, 62 (2015), 660-664.

(3) An article by E. Klarreich in Quanta Magazine on Zhang's discovery with extensive quotes from Andrew Granville, Dan Goldston, and Yitang Zhang. It discusses the work of GPY and includes intuitive descriptions of the level of distribution of the primes and sieving: Unheralded mathematician bridges the prime gap, Quanta Magazine, May 19th 2013 .

(4) An article by M. Segal in Nautilus Magazine also on Zhang's discovery quoting Klarreich (3) and reporting on an extensive interview with Zhang. It also states that Zhang's discovery gave rise to 100 news items in the public press in May 2013: The twin primes hero, Nautilus Magazine, September 2013.

(5) A web page by Prof T. T. Moh, Zhang's PhD supervisor at Purdue University, giving his recollections of Zhang and their interactions: Zhang, Yitang's life at Purdue (Jan 1985-1991), Aug 2013 revised in boldface 2018.

(6) An article by Alec Wilkinson published in the New Yorker entitled The pursuit of beauty: Yitang Zhang solves a pure-math mystery, New Yorker magazine, 2nd February 2015, giving many quotes from Zhang and lots of background information.

(7) Observations of János Pintz on the contributions he and Motohashi made to the work of and Zhang, in particular on the difficulty they had interacting with Zhang concerning the significance of their work for his results: Bull. London Math. Soc. 40 (2008), 298-310. See the Concluding remark page 309. See also in this regard the remark of Terry Tao following the statement of Theorem 3 in his article Notes on Zhang's prime gaps paper, June 2013..

(8) Zhang Backstory timeline:

1970's: Zhang worked on a collective farm in China.

Late 1970's - early 1980s: Zhang entered Bejing University and graduated with a masters degree in 1984.

1985-1991: Zhang undertook graduate studies at Purdue University and wrote a thesis on the Jacobian Conjecture under Prof. T. T. Moh.

1990's: Zhang continued to think about mathematics but supported himself without a position in the subject.

2000: Zhang accepted a position as an adjunct professor at the University of New Hampshire.

2008: Working at first independently and then together Motohashi and Pintz published their famous paper "A smoothed GPY sieve", indicating how the introduction of smoothness would enable bounded prime gaps to be derived.

Mid 2000 - early 2010's: Zhang considered the works of Goldston, Motohashi, Pintz and Yildirim, as well as a range of other advanced number theory methods including those of Bombieri, Fouvry Iwaniec, and Linnik.

July 3rd 2012: on visiting the Colorado family farm of Jacob Chi (Music Director of the Pueblo Symphony), Zhang gets his final idea on how to complete the proof of his breakthrough theorem that primes have bounded gaps.

April 2013: Zhang submits a manuscript to the Annals of Mathematics.

Late 2013: Zhang's paper "Bounded gaps between primes" is published in ultra rapid time.

2013-2014: Terence Tao formed a Polymath project (called Polymath8a), deeply analysing the GMPY/Zhang method, introducing new concepts (e.g. dense divisibility) and very significantly improving the result by lowering the bound from 7x10^7 to 4680.

Maynard, Tao and Polymath8

(1) A book by Vicky Neal addressed to a general audience with at least school level mathematics knowledge, but containing a lot of interesting information on how some of the principal contributors came to their results and interacted with each other: "Closing the gaps: the quest to understand prime numbers", Oxford, 2017.

(2) An article in the Newsletter of the European Mathematical Society by D. H. J. Polymath which looked back over the work of Polymath8, with named observations from Terence Tao, Andrew Gibson, Pace Nielsen, James Maynard, Gergely Harcos, David Roberts, Andrew Sutherland, Wouter Castryck, Emmanuel Kowalski, and Philippe Michel: "The bounded gaps between primes" Polymath Project, EMS Newsletter December 2014, p13-23.

(3) The link to the Polymath8 home page: this provides a very rich source of information on the work of the project and related matters. It contains detailed tables, errata for some related articles, references and many hundreds of links. A lot of this is technical, but also extremely valuable for finding leads for the backstory.

(4) Maynard Backstory timeline:

2009: Maynard completed a masters degree at Cambridge University.

2013-2014: Maynard completed a PhD at Oxford University supervised by Roger Heath-Brown and then took up a one year post doctoral fellowship with Andrew Granville at the University of Montreal. He studied the works of Goldston, Motohashi, Pintz, Yildirim and Zhang and developed his own ideas based on earlier multi-divisor sieving ideas of Selberg. He kept in touch with Polymath8, but continued along his own path.

19th November 2013: Simultaneously and independently Terry Tao was also attacking the bounded gaps problem using multi-divisors, but using Fourier analysis rather than the more combinatorial approach of Maynard. Granville, who knew both mathematicians well, assisted both to obtain acceptable publication, James on ArXiv and then in the Annals of Mathematics, and Terry, first on his blog site and then as part of Polymath8b's published article. As it turned out Maynard lowered the bound to 600 (actually his method gives a somewhat better result) and Polymath8b to 246, a bound which has not thus far been improved. Polymath8b made a very comprehensive contribution to the field, in particular going well beyond the computational setting of Maynard and deriving limits to what the Maynard/Tao method could achieve.

2015: In his Annals paper Maynard does quite a lot else besides significantly improving the prime gap size. For example he derives bounded gaps for any specified number of primes (as did Polymath8b) and positive relative density results for primes in tuples.

(5) Polymath8 Backstory timeline:

May 2013: Zhang published a preprint of his bounded gaps paper containing the tantalizing phrase "this result is not optimal", the result being there are an infinite number of consecutive primes distance apart less than 70,000,000.

Following this announcement, over the space of days first Lewko, then Trudgian, then Morrison published on the web successive small improvements to Zhang's bound, by making small adjustments to his argument.

In early June 2013 a project named Polymath8 was formed with the aim of optimizing Zhang's argument as far as possible. After just 3 months of concentrated collaborative work the bound was lowered to 4680 and it was decided to write up the work for publication.

The write up was almost finished when the news that Maynard had used a new method to both simplify the proof and reduce the bound to 600. The group then shifted its attention to this new approach, which had many ingredients in common with one developed by Terry Tao. The new project was named Polymath8b with the earlier work renamed Polymath8a. With another 8 months of concentrated collaborative work the bound was lowered to 246, with the write up published in 2014 in the journal "Polymath research in the mathematical sciences".