Stanley Yao Xiao, Department of Pure Mathematics, University of Waterloo
“An Overview of the Bounded Gaps Between Primes Problem”
It is a long standing conjecture, since antiquity, that there exist infinitely many consecutive prime numbers that are separated by 2, which is of course the closest possible distance. The prime number theorem shows that the gap between pn and pn+1 is on average log pn. It is surprising then that even proving the existence of infinitely many gaps smaller than some constant multiple of the average has proved difficult for the century that ensued the proof of the prime number theorem. In a breakthrough paper in 2009, Goldston, Pintz, and Yildirim proved that for any small constant c > 0, there exist infinitely many primes p, q such that |p − q| < c log p. In doing so they were able to relate the gaps between prime problems with a famous conjecture of Elliott and Halberstam, the first time anyone was able to connect the bounded gap problem to a major and fundamental conjecture in number theory. In May 2013, Yitang Zhang announced a proof of the existence of bounded gaps between primes and just six months later, James Maynard gave a drastically different and technically innovative proof which led to far superior estimates on the size of the gaps. In this talk I will give a brief outline of the arguments of GPY, Zhang, and Maynard.