A few days ago, I met a bright student, Nia, who had two burning desires: to understand the distribution of prime numbers and to run an ultramarathon. She made a running plan for herself with the following rule: she would start with the first prime number, 2, and on each day, would run as many kilometres as needed to reach the next prime.
Prime numbers, that is, positive integers with no factors other than 1 or themselves, are considered the atoms of number theory: multiplied among themselves in all combinations, they “make up” all the integers. These numbers have intrigued mathematicians over several centuries. The patterns in which they appear in the number system have still not been properly understood (and one can't even say that there is no pattern at all). By looking at the first few primes, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, etc, Nia saw that the gaps between consecutive primes seemed to grow steadily up to a gap of 6 before tapering down to 2. This pattern sort of continued for a while until Day 30, when, after a steady practice of 2 - 6 km every day, she suddenly found herself running 14 km, from 113 to 127! For several days thereafter, she would mostly run small distances with occasional spikes of up to 20 km. But, on Day 217, the distance suddenly increased to 34 km from a maximum of 20 kilometres in all her previous runs. Starting to doubt her prime-gaps-based running plan, she asked the following questions:
(1) Bothered with the frequency of 2 km days, she wondered if she would ever reach a stage beyond which all prime gaps would be more than 2. A famous conjecture in number theory, called the twin prime conjecture, predicts that the answer to this question is no; that is, there are infinitely many pairs of primes with gap 2. So, according to this conjecture, even if she were to run forever, 2-km-gaps would keep appearing!
(2) So far, the prime gaps seemed to be growing manageably on most days, but sudden gaps, much larger than expected, would make it difficult for her to cope. So, up to any finite number M, she wondered if the largest gap between consecutive primes would be “much larger” than the average gap between them.
Both the above questions are deep and longstanding problems in number theory. Remarkable progress has been made towards answering these questions in the last few years.
In April 2013, an unassuming mathematics lecturer at the University of New Hampshire in USA by the name of Yitang Zhang announced that there is a number N less than 70 million such that there are infinitely many pairs of consecutive primes with gap N. Strictly speaking, this doesn’t do much for Nia's running plan as she wants to know if this is true for really small values of N, like 2. But, this was the first time that a mathematician could claim (and correctly prove) the existence of a finite N with the above property. What this means is that as far as you go on the number line, you will keep encountering pairs of consecutive primes that differ by this bounded gap, N. Immediately after Zhang announced his work, some mathematicians announced the Polymath8 project, an international collaborative effort to improve Zhang’s bound. This bound has been brought down from 70 million to 246, independently by James Maynard and Terence Tao. So, while the twin prime conjecture still remains unresolved, showing that there are infinitely many pairs of primes with bounded gap is a breathtaking development, certainly among the most promising discoveries of the 21st century!
Nia's second question, which can be viewed as a reverse of her first, also has an interesting history. Paul Erdos, one of the most productive and famous mathematicians of the 20th century, often posed number-theoretic problems to his contemporaries. He would offer small cash prizes for correct solutions. However, about 80 years ago, he offered a huge prize of $10,000 to anyone who would solve what he called the prime gaps conjecture. This is a slightly technical conjecture, which predicts a lower bound for the largest gap between consecutive primes up to a number x. This conjecture was proved in August 2014, by Kevin Ford, Ben Green, Sergei Konyagin and Terence Tao and independently by James Maynard. Clearly, Tao and Maynard were inspired by their previous work on small gaps between primes.
Paul Erdos is no longer around to reward these people, but the mathematical community feels that this is not the last word on this subject and the bound can be further improved. So, Terence Tao has offered another $10,000 for a better bound!
This answers Nia's second question in the affirmative. Thus, if she were to strictly follow her prime-based running plan, it would be futile to expect a steady growth of gaps between consecutive primes. Every once in a while, she would encounter a prime gap far bigger than what she had seen before.
Last I heard, Nia has turned to more traditional running plans for her marathon training. However, her interest in primes continues. She is working towards Tao's open problem and hopes to win the $10,000 offered by him!
Kaneenika Sinha, IISER, Pune