There are math problems that are thoroughly incomprehensible to the layman – the Riemann hypothesis needs a fair amount of sophisticated math to explain, for example. And then there are those that even a ten-year-old could understand. The famous Twin Prime Conjecture (footnote 1) is definitely in the latter category – conjectured by Alphonse de Polignac in the 19th century, it states that there are infinitely many pairs of prime numbers that differ by two – hence the name "twin" prime. For example, 3 and 5 are twin primes, and so are 71 and 73. As we get to larger numbers, prime numbers show up less frequently. However, we have also found ridiculously big twin primes [1] – the current record being 2996863034895 × 21290000 – 1 and 2996863034895 × 21290000 + 1, with 388,342 decimal digits – which leads us to believe that there might be an infinite number of twin primes; no matter how large you go, you might always expect a pair of twin primes ahead.

There isn't very much to say about the problem, but more interesting is how progress has been made in recent years. People might think of math research as a solitary affair, imagining a mathematician locking himself / herself up in a room until he / she find a solution. And certainly there are two remarkable people involved in this story – Yitang ZHANG and James MAYNARD – but at the heart of a lot of progress sits a huge collaborative project called the Polymath Project.

Yitang ZHANG's life story is certainly worthy of a movie. He was born in China, and was sent to a labour camp with his mother during the Cultural Revolution, which interrupted his education for about eight years [2]. He later went to the US for his PhD study, but parted ways extremely unhappily with his supervisor. Unable to find a job in academia after graduation, he worked a number of odd jobs for seven years, including a stint as an accountant at his friend’s Subway franchise, before becoming a lecturer at the University of New Hampshire in 1999 [2]. No one had heard of him before his breakthrough, which makes his pioneering work even more remarkable. Before Zhang’s breakthrough and subsequent work built on that by others, there was no real upper bound on the maximum gap between primes; efforts were, of course, made on the problem, but it largely refused to budge. In April 2013, Zhang published his results, showing that there were infinitely many primes with a gap less than 70 million – a large number, but a finite one. This was news that set the math community ablaze. Amidst a flurry of interest in the new result, Polymath8 began.

The Polymath Project was founded by mathematician Timothy GOWERS, who started the first project on his blog in 2009 [3]. It is a massive mathematical collaborative project, with a goal set out at the beginning, and different mathematicians contributing to the goal through online discussions open to everyone. Usually a mathematician serves as a host, with their blog becoming the project’s discussion forum, and allowing all people to participate as long as they could chip in with some insight [4]. The results were usually published in academic journals under the pseudonym D H J Polymath. Polymath8, the eighth project in the series, was started by Terence TAO, arguably the world’s most famous mathematician, in June 2013 [5]. The project set out to improve Zhang's work and give a more accurate upper bound of the prime gap. A good number of mathematicians joined in, and the project generated a lot of excitement. The bound was reduced every day, eventually falling to 4680 in July 2013, where the project momentarily came to an end [6].

In a dramatic development, James MAYNARD, now a professor at Oxford, published another result in November 2013. Fresh off a PhD, he had been working independently of Zhang and the Polymath Project, but gave a bound of 600 using an entirely different method [7]. Had he published his result earlier, it would have been him who made the headlines – 600, after all, is a far more headline-worthy bound than 70 million. Maynard's result was a better bound than that from the Polymath Project, and soon the project's participants turned their attention to Maynard's new methods, hoping that the conjecture could be solved once and for all.

Thus began Polymath8b, a continuation of the previous project, which now involved James MAYNARD; together, the mathematicians tried to push the gap down further, but this time they were met with other forms of resistance. After all, one could only improve bounds for so long – nevertheless, the results were quite encouraging. In April 2014, just one year after Zhang published his results, the bound stood at 246; assuming an additional result on the distribution of primes, known as the Elliott-Halberstam conjecture, the bound can be reduced to 6 (footnote 2) [8, 9]. However, to lower the bound to 2 for the Twin Prime Conjecture, new methods have to be invented in order for further progress.

The Polymath Project has always been an interesting exposition in mathematical research – the open-sourced nature of the project causes results to come extremely quickly, at a speed far greater than regular mathematical collaborations, which often involve just a few mathematicians corresponding with each other. It is often likened to drinking out of a fire hose – the speed at which breakthroughs happen is simply thrilling. On the flip side, some may be hesitant to participate, as all their mistakes will be left open on the discussion forum permanently; others argue that the speed does not allow mathematicians to dwell on some things for a longer period of time, perhaps yielding more results [10]. The question is, in this age of the internet, are Polymath Projects the way forward in mathematical research?

Perhaps time will tell. The Polymath projects so far have tackled tasks that were easily divided into parts, so different people could work on different components before piecing the details together. But if there is one moral from the story of the Twin Prime Conjecture, it is that breakthroughs come in different forms, from a lone mathematician working alone for years, to the rapid-fire breakthroughs of a group think tank. There are still numerous mathematical questions waiting for us to solve, and you, dear reader, might solve one of those some day!

Interested and mathematically advanced readers may wish to read Terence TAO’s blog (but be warned that you will need a lot of mathematical background!): https://terrytao.wordpress.com/

Timothy GOWERS, the founder of Polymath, also has a blog, which is more accessible to the layman: https://gowers.wordpress.com/

James MAYNARD, one of the main contributors to this problem, also talks about the Twin Prime Conjecture on Numberphile: https://youtu.be/QKHKD8bRAro

As is now customary for these articles, we will end with a problem, this time about prime numbers [11]:

Let n be an integer ≥ 4.
Our goal is to arrange the numbers 1, 2, …, n in a circle, so that any two neighboring numbers add up to a prime number. For example, (1, 2, 3, 4) is a valid arrangement for n = 4:

However, this proves to be impossible for odd n. Why?
(As a bonus question, if n + 1 and n + 3 are twin primes, then can you construct an arrangement for n?)
Solution: http://sciencefocus.ust.hk/solution_020

#### Footnote:

1. Conjecture: A proposed statement waiting to be proved. A proved conjecture becomes a theorem.
2. A small aside: primes that have a difference of six are, somewhat hilariously, known as "sexy primes" ("sex" is the Latin prefix for "six", so the label is as in twin primes).

#### References:

1. Caldwell, C. K. (2021). The Largest Known Primes -- A Summary. Retrieved from https://primes.utm.edu/largest.html
2. Wilkinson, A. (2015, February 2). The Pursuit of Beauty. The New Yorker. Retrieved from https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
3. Gowers, W. T. (2009, March 24). Can Polymath be scaled up [Web log post]? Retrieved from https://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/
4. Kalai, G. (2021, January 12). Proposals for polymath projects. Retrieved from https://mathoverflow.net/questions/219638/proposals-for-polymath-projects
5. Tao, T. (2014). Polymath8 | What’s New [Web log post]. Retrieved from https://terrytao.wordpress.com/tag/polymath8/
6. Polymath, D. H. J. (2014). New equidistribution estimates of Zhang type. Algebra & Number Theory, 8(9), 2067-2199. doi:10.2140/ant.2014.8.2067
7. Maynard, J. (2019). Gaps between primes. ArXiv. arXiv:1910.13450 [math.NT]
8. Polymath, D. H. J. (2014). Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences, 1. doi:10.1186/s40687-014-0012-7
9. Polymath Project. (2014). Bounded gaps between primes - Polymath Wiki. Retrieved from https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes
10. Polymath, D. H. J. (2014). The "bounded gaps between primes" Polymath project - a retrospective. ArXiv. arXiv:1409.8361 [math.HO]
11. Massachusetts Institute of Technology. (2020). PRIMES Math Problem Set: Solutions. Retrieved from https://math.mit.edu/research/highschool/primes/materials/2020/entpro20sol.pdf

Author:
Sonia Choy
Student Editor, Science Focus
The Hong Kong University of Science and Technology

Design:
Tsang Cheuk Hei
Graphic Designer, Science Focus
The Hong Kong University of Science and Technology

January 2022