## Leonhard Euler

Eda Orakçı

For this month’s edition of The Math Chronicles, I wanted to focus on a documentary. This format may prove to be a change of pace, as the film focuses on the daily life of a mathematician who was unheard of until recently, and may still be unheard of today outside of mathematical circles. Complete with interviews from journalists, mathematicians and family friends, the documentary paints the picture of a man who has made a breakthrough contribution to Number Theory. His life, his marriage, his emigration from China, and the subsequent challenges he has faced in academia. His irregular job at Subway and his barely noticeable presence in the international mathematical community. His breathtaking paper submitted to Annals of Mathematics sending shockwaves in prestigious academia and taking the lid off of mathematical discovery as it relates to prime numbers.

The documentary is Counting From Infinity, published in 2015. The mathematician in question is Yitang ‘Tom’ Zhang. This name may be familiar to many in this publication, but if not, I’m assuming the Twin Prime Conjecture is. Alas, here’s a small explanation: The twin prime conjecture is the assertion that there are infinitely many prime numbers that differ by two. The concept itself can be concretely traced back to Alphonse de Polignac in late 19th Century; however, it is believed that the conjecture may date back to Ancient Greece. Up until the year 2013, progress made towards proving this conjecture was small. This is the moment Zhang enters the story.

He was born in 1955 in China, but moved to the United States at age 30. The film regularly shines light on his childhood, and his family’s struggles in the newly established communist regime. Due to espionage suspicions, Zhang was banned from highschool and explored his blooming interest in mathematics in relative isolation. After the Cultural Revolution, he was given the opportunity to attend college, and he accepted the offer in glee, enrolling in Peking University for an undergraduate degree in mathematics. His brilliance was noticed by his teachers, and he swiftly climbed up the ranks to find himself in Purdue University working on his doctorate. These strings of academic achievement would be short-lived, however, as his advisor refused to write him a letter of recommendation upon graduation, and Zhang found himself unemployed. He would return to academia eight years later as a lecturer at the University of New Hampshire. Although closer to his true passion of mathematics, he was effectively excluded from higher mathematical circles.

That’s why his paper is all the more surprising. Taking inspiration from the GPY theorem proven by Daniel Goldston, Janos Pintz, and Cem Yıldırım in 2005, Zhang had found a way to prove that there are infinitely many pairs of prime numbers that differ by 70,000,000 or less. This number may seem a long way from 2, but it must be kept in mind that as numbers increase, the average distance between prime numbers becomes increasingly larger. In fact, the average distance between prime numbers can be estimated as log(n) where there are a total of n consecutive numbers starting from 0 and increasing by 1. It was of great importance that it was proven possible for a relatively small gap and increased attention and determination in the international mathematical community on making progress towards proving the Twin Prime Conjecture.

Figure 1: The now legendary paper published in Annals of Mathematics in 2013 by Yitang Zhang.

The film details the aftermath of his discovery, and how it has affected Zhang’s life. With a newfound spotlight of fame now upon him, he received numerous awards and prizes over the course of the film by esteemed institutions. The film features an interview with Zhang stretching for the entire runtime of 1 hour. The film also focuses on the flurry of new discoveries that has followed that of Zhang’s.

In the online forum Polymath, a mathematical project called the “Bounded Gaps Between Primes” was coordinated. As mathematicians found new ways of decreasing the upper bound of the gap between two prime numbers stretching to infinity, the site updated, causing more excitement and an optimistic outlook on progress. In fact, the film provides another keystone interview. This time it is with Oxford mathematician James Maynard, who just a few months after the publication of Zhang’s work, published an alternative proof for the bounded gaps of prime numbers and decreased the upper bound even further. As it stands today, this upper bound has been lowered to 246 with combination and improvements of both Zhang’s and Maynard’s work.

The film is detailed and revealing; however, it is rather slower-paced with long shots of Zhang occasionally accompanied by his wife walking along gardens where he feels inspired. I would recommend the film for anyone interested in the personal life of a mathematician who has accomplished greatness, and to take a closer look at the early life, work, and current life of someone who had the odds stacked up against them. Genius comes in many forms, and the idea that a person with a ‘regular’ life is capable of achieving it through determination and perseverance may provide inspiration to anyone looking for it. If that sounds like you, you have found the film to watch for tonight. If not, there are hours upon hours of lectures on prime numbers online, even one given out by Zhang himself. The content is there, and it is accessible, and it is possible to understand the mathematics behind this discovery with (admittedly a lot of) digging.

Here’s to all new discoveries of tomorrow.

Figure 2: Yitang Zhang explaining his proof in a lecture entitled “Bounded Gaps Between Primes” in 2014.

Figure 3: “Don’t forget the zero!”

References

[1] Counting From Infinity: Yitang Zhang And The Twin Prime Conjecture. Directed by George Csicsery, 2013.

[2] Numberphile, “Gaps between Primes”, 2013. Picture 1 taken from this source.

https://www.youtube.com/watch?v=vkMXdShDdtY&ab_channel=NumberphileNumberphileVerified

[3] Yitang Zhang, “Bounded Gaps Between Primes”, 2014. Video by Stony Brook Mathematics Colloquium. Picture 2 & 3 taken from this source.

https://www.youtube.com/watch?v=Z5zvhqyO7IM&ab_channel=GraduateMathematics