## A Good Will Hunting Story from China

### Video

There is an surprising and interesting mathematical story from China. At the center of the story is a migrant worker in China named Yu Jianchun who is generating excitement and awe because of his work in complex and esoteric math problems despite having received no university education, much less formal mathematical training. Facts about the story are still scanty. But he is attracting a lot of attention from academics and from the general public in China and internationally. Here is an article from CNN about Yu. Many people are seeing a parallel between Yu and the character Will Hunting in the Oscar-winning movie “Good Will Hunting”.

After studying at a vocational school, Yu Jianchun became a migrant worker going from place to place working as a parcel delivery man. He always has a passion for mathematics and he spends almost all of his spare time studying it. He is also persistent (and probably stubborn as well). He has spent 8 years working on the problem that currently garners him national and international attention. Whenever he found work in a new city, he always seek out the mathematics professors at the local university in hope of finding confirmation for his math work. He was ignored until a math professor at Zhejiang University, Cai Tianxin, invited Yu in June 2016 to present his math work at a seminar.

The math work of Yu that is generating buzz involves Carmichael numbers, which are odd integers that are prime-like (the usual term is pseudoprimes). For both theoretical and practical reasons, it is critical to test whether a given large odd integer is a prime number. Carmichael numbers are integers that are not prime but yet pass the Fermat’s test for prime numbers. So being able to weed out the Carmichael numbers from the prime numbers will be critical. Testing whether a number is a prime number is more than an intellectual curiosity. Prime numbers are the back bones of encryption systems such as the RSA algorithm, which makes online shopping safe and secure. So the study of prime numbers and Carmichael numbers has implications for information security.

Carmichael numbers are named after R. D. Carmichael who discovered 15 such numbers in 1910. These numbers are very rare. For example, of all the numbers that are less than one billion, there are only 646 Carmichael numbers. In contrast, there are 50,847,534 (over 50 millions) prime numbers below one billion. R. D. Carmichael conjectured that the number of Carmichael numbers is infinite. It was finally proven in 1994 that there are infinitely many such numbers, i.e. there is no upper bound on Carmichael numbers. No matter how big the whole number $n$, it is proven that there is always a Carmichael number larger than $n$.

What is special about Yu’s work is that he has discovered a new way to identify Carmichael numbers that is different from the classic algorithm. However, there has not been any precise mathematical statement on what Yu’s results are. So it is hard to get a sense of how special or how significant his results are. In the CNN article referenced above, William Banks, a mathematician who works with Carmichael numbers, seemed to indicate that Yu’s results are about formulas that can generate Carmichael numbers. Yet the same CNN article also indicated that Yu’s work is about an alternative method to verify Carmichael numbers. Is it a new formula for generating Carmichael numbers or is it a new test to check whether a given number is a Carmichael number?

It is likely that Yu’s results are currently being verified. Cai Tianxin, the math professor at Zhejiang University who invited Yu to give a talk, plans to publish Yu’s theory in a book on Carmichael numbers. So in time we will have a better sense of Yu’s achievement.

Another refreshing point about the story of Yu Jianchun is that Carmichael numbers are in the news! Occasionally prime numbers are in the news. For example, when someone finds a new largest prime number or when someone proves a long standing problem about prime numbers. But rarely do we see Carmichael numbers being mentioned by the major news outlets.

It is hard for me to comment on Yu’s work on Carmichael numbers since details are not available. However, Carmichael numbers are an interesting concept in number theory. Here’s an introduction to Carmichael numbers. Here’s an article on how to use Fermat’s test for prime numbers. Here’s another discussion on Carmichael numbers. Finally, here’s another discussion on Fermat’s test.