Page 24 - Science Focus (issue20)

waiting for us to solve, and you, dear reader, might
The Twin Prime
Conjecture and
the Polymath Project
孿生質數猜想與博學者計劃
By Sonia Choy 蔡蒨珩
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
There are math problems that are thoroughly to budge. In April 2013, Zhang published his results,
incomprehensible to the layman – the Riemann showing that there were infinitely many primes with
hypothesis needs a fair amount of sophisticated a gap less than 70 million – a large number, but a
math to explain, for example. And then there are finite one. This was news that set the math community
those that even a ten-year-old could understand. ablaze. Amidst a flurry of interest in the new result,
The famous Twin Prime Conjecture (footnote 1) is Polymath8 began.
definitely in the latter category – conjectured by
Alphonse de Polignac in the 19th century, it states T h e Po l y m ath Pro j ect wa s fo u n d ed by
that there are infinitely many pairs of prime numbers mathematician Timothy Gowers, who started the
that differ by two – hence the name "twin" prime. first project on his blog in 2009 [3]. It is a massive
For example, 3 and 5 are twin primes, and so are 71 mathematical collaborative project, with a goal set
and 73. As we get to larger numbers, prime numbers out at the beginning, and different mathematicians
show up less frequently. However, we have also found contributing to the goal through online discussions
ridiculously big twin primes [1] – the current record open to everyone. Usually a mathematician
being 2996863034895 × 2 1290000 – 1 and 2996863034895 serves as a host, with their blog becoming the
× 2 1290000 + 1, with 388,342 decimal digits – which leads project’s discussion forum, and allowing all people
us to believe that there might be an infinite number of to participate as long as they could chip in with
twin primes; no matter how large you go, you might some insight [4]. The results were usually published
always expect a pair of twin primes ahead. in academic journals under the pseudonym D.H.J.
Polymath. Polymath8, the eighth project in the series,
There isn't very much to say about the problem, was started by Terence Tao, arguably the world’s
but more interesting is how progress has been made most famous mathematician, in June 2013 [5]. The
in recent years. People might think of math research project set out to improve Zhang's work and give a
as a solitary affair, imagining a mathematician more accurate upper bound of the prime gap. A
locking himself/herself up in a room until they find good number of mathematicians joined in, and the
a solution. And certainly there are two remarkable project generated a lot of excitement. The bound was
people involved in this story – Yitang Zhang and reduced every day, eventually falling to 4680 in July
James Maynard – but at the heart of a lot of progress 2013, where the project momentarily came to an end [6].
sits a huge collaborative project called the Polymath
Project. In a dramatic development, James Maynard,
now a professor at Oxford, published another result in
Yitang Zhang's life story is certainly worthy of a November 2013. Fresh off a PhD, he had been working
movie. He was born in China, and was sent to a labor independently of Zhang and the Polymath Project,
camp with his mother during the Cultural Revolution, but gave a bound of 600 using an entirely different
which interrupted his education for about eight years method [7]. Had he published his result earlier, it
[2]. He later went to graduate school in the U.S., but would have been him who made the headlines – 600,
parted ways extremely unhappily with his supervisor. after all, is a far more headline-worthy bound than
Unable to find a job in academia after graduation, 70 million. Maynard's result was a better bound than
he worked a number of odd jobs for seven years, that from the Polymath Project, and soon the project's
participants turned their attention to Maynard's new 1 Conjecture: A proposed statement waiting to be proved. A
solve one of those some day!
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;
proved conjecture becomes a theorem.
together, the mathematicians tried to push the gap
2 A small aside: primes that have a difference of six are, somewhat
forms of resistance. After all, one could only improve
so the label is as in twin primes).
down further, but this time they were met with other hilariously, known as “sexy primes” (“sex” is the Latin prefix for “six”,
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 As is now customary for these articles, we will end with
a problem, this time about prime numbers [11]:
can be reduced to 6 (footnote 2) [8, 9]. However, to Let n be an integer ≥ 4.
lower the bound to 2 for the Twin Prime Conjecture,
new methods have to be invented in order for further Our goal is to arrange the numbers 1, 2, …, n in a
circle, so that any two neighboring numbers add up
progress. to a prime number. For example, (1, 2, 3, 4) is a valid
arrangement for n = 4:
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 doesn’t allow mathematicians to dwell
on some things for a longer period of time, perhaps However, this proves to be impossible for odd n. Why?
yielding more results [10]. The question is, in this age of (As a bonus question, if n + 1 and n + 3 are twin
the internet, are Polymath Projects the way forward in primes, then can you construct an arrangement for
mathematical research? n?)
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 Solution 答案
years, to the rapid-fire breakthroughs of a group think
tank. There are still numerous mathematical questions
Interested and mathematically Timothy Gowers, the founder of James Maynard, one of the main
advanced readers may wish to Polymath, also has a blog, which contr ibutors to this problem,
read Terence Tao’s blog (but be is more accessible to the layman: also talks about the Twin Prime
warned that you will need a lot of 博學者計劃的創辦人Timothy​Gowers Conjecture on Number phile:
mathematical background!): 也有寫網誌，裡面有不少比較容易看懂的 為問題帶來重大突破的​James​Maynard
如果你對以上的數論有興趣，又學過 內容： 也有在著名Youtube頻Numberphile上
一點大學數學的話，可以看一看陶哲 https://gowers.wordpress.com/ 和主持討論孿生質數猜想：
軒的網誌（但是他寫的東西比較深，​ https://youtu.be/QKHKD8bRAro
看不懂也不要緊！）：
https://terrytao.wordpress.com/
Terence Tao’s blog Timothy Gowers’ blog The Numberphile video on
陶哲軒的網誌 Timothy Gowers的網誌 the twin prime conjecture
Numberphile的孿生質數猜想影片
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