Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field.

‘The author has gathered almost 100 year's worth of progress on this family of problems into one volume, and this alone will be very helpful to anyone pursuing research in the field. Recommended.’

M. Bona Source: Choice

‘a wonderful tale of how two lesser-known mathematicians worked extremely hard to solve an intriguing, long-standing open problem that so many leading experts could not.’

Sam Chow Source: London Mathematical Society