# How “small” can the gap be? -A huge landmark in number theory.

Yitang Zhang, lecturer in mathematics at the University of New Hampshire
Photo courtesy of Lisa Nugent/UNH Photographic Services

What’s the gap between consecutive primes? One can easily observe that the gap will keep increasing as the primes become far rarer. However, for the bounded gap, there exists infinitely many pairs of primes. This is a form of the twin prime conjecture.

Twin prime conjecture has been one of the most notorious questions in number theory for over 160 years, and the topics around this conjecture is never off the table. The proof of the conjecture seems extremely difficult. Even though the proof of the weak form is very hard. The content of the twin prime conjecture, which states: There are infinitely many primes p such that p+2 is also prime. In 1849, Alphonse de Polignac made a more general version of twin prime conjecture which can be interpreted, and extended by Terence Tao in the blog posted on June 3, 2013: Suppose one is given a $k_0$tuple $\mathcal H = (h_1, h_2, \cdots, h_{k_0})$ of $k_0$ distinct integers for some $k_0\geq 1$, which is arranged in increasing order. When is it possible to find infinitely many translates $n+\mathcal H$ of $\mathcal H$ consisting entirely of primes?
The case $k_0=1$ is is Euclid’s theorem, famous and one of very first theorems. However, on infinitude of primes, the case $k_0=2$ is the twin prime conjecture.

On April 17, 2013, Yitang(“Tom”) Zhang, a very inactive mathematician at the University of New Hampshire, announced his breakthrough proof of “bounded gaps” conjecture (a weak form of the twin primes conjecture) about the distribution of prime numbers–a milestone result to the weak form of twin primes conjecture. Zhang did a very impressive work by proving the theorem,

Theorem (Bounded gaps between primes) There exists a natural number $H$such that there are infinitely many pairs of distinct primes $p, q$ with $| p-q|.

This theorem is implied by Motohashi-Pintz and Zhang conjecture, a specialized form of the Elliott-Halberstam conjecture. Thanks to the previous works(but in a much simpler way) did by Fouvry-Iwaniec, Friedlander-Iwaniec and Bombieri-Friedlander-Iwaniec, Zhang showed that the conjecture is true for all $0<\varpi<1/1168$, where $\varpi$ is a parameter in Motohashi-Pintz-Zhang conjecture. Via Dickson-Hardy-Littlewood(DHL) conjecture, Zhang obtained $k_0 = 3,500,000$. Then, the gap of infinitely many pairs of consecutive primes is 70,000,000. Before that most people believed the gap is $\infty$.

Just like Henryk Iwaniec’s comments on Zhang’s work, the proof is elegant and breakthrough, but the gap will be narrowed very soon. Through recent efforts of Scott Morrison, Terence Tao and other mathematicians, the newest bound gap record is 10,206. See this web page for the latest bounds.

Through many people’s eye, Zhang undoubtedly looks like a parvenu who gained fame in one night. After obtaining the PhD degree from Purdue University in 1991, Zhang worked for several years as an accountant, a delivery worker for a New York City restaurant, in a motel in Kentucky and in a Subway sandwich shop before working as a lecturer at University of New Hampshire. Moreover, one can only found two papers written by him in the Zentralblatt MATH database. All these facts seems quite against the normal routine for a good mathematician. Learning from some interviews, we find out Zhang is such a guy who doesn’t care about the reputation and tenures. And he never try to write a paper in order to get a job in the top schools. The only thing matters to him is math itself. Zhang’s success can be concluded by, “it’s not that I’m so smart, it’s just that I stay with problems longer.” (as cited by Albert Einstein)

Here comes the question. Would you like to choose a long-term and “risky” project such as Fermat’s Last Theorem which took Andrew Wiles 9 years to prove that, or you would pick a bunch of short-term ones that can help you seek a neat job in the future?