1 - Introduction

Summary

In 2013–14 Zhang, Polymath8, Maynard, Tao, both separately and together, showed that the gap between an infinite number of consecutive primes was less than 70 million, and then lowered the upper bound to 246. Progress then ceased. This chapter gives contextual and introductory material needed to derive the best results. Section 1.3 has an overview of the book. Section 1.4 describes Timothy Gowers’ idea of a polymath project, and lists contributors to Polymath8. Section 1.5 gives a time-line of the developments, Section 1.6 discusses the twin primes constant and the Dickson–Hardy–Littlewood conjecture, Section 1.7 delves into the nature of the prime gap distribution by discussing the issue of which prime gap is most common, and reports on recent work on “jumping champions”, Section 1.8 gives the derivation of some useful properties of the von Mangoldt function, Section1.9 discusses the Bombieri–Vinogradov theorem, Section 1.10 introduces admissible tuples, which describe patterns of primes which are expected to repeat infinitely often, and derives the intriguing relationship between the Dickson–Hardy–Littlewood conjecture and the second Hardy–Littlewood conjecture. Section 1.11 gives a brief guide to the literature and reader’s guide.In an end note, there is a summary table for results on large gaps between consecutive primes.