This chapter gives a sample of work done on prime gaps spanning more than half of the twentieth century, showing the importance of the problem and giving a context for what was to follow. Erdős proved in 1940 that there are an infinite number of consecutive primes strictly less than the average gap. His proof is given in Section 3.3. Section 3.4 gives the theorem of Bombieri and Davenport showing that there are an infinite number of primes less than half the average gap. Their proof uses many advanced methods and tools with a fundamental lemma of 10 steps. Section 3.5 gives Granville and Soundararajan’s version of the work of Maier, which has applications to primes in intervals as well as prime gaps.

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.

This appendix gives 1000 integer shifts sufficient to regenerate quickly 1000 admissible tuples. These tuples are "inductive" in the sense that for each integer k the k'th initial segment of the regenerated integers is an admissible tuple and the k+1'st is the smallest with one additional element which is admissible.

It is fascinating to see that while Polymath8a was making improvements to Zhang’s method, James Maynard and Terence Tao, independently using combinatorial and analytic methods respectively, were using a completely different approach to study bounded gaps. This was based on an idea suggested by Selberg, and is called the multidimensional sieve. Tao was later to incorporate his method into the work of Polymath8b reported in Chapter 8, while Maynard’s work is given in this chapter. Three sections are devoted to developing properties of the sieve. Then a simplified form of the derivation of an essential integral formula is given. After detailing Maynard’s optimization procedure, and his Rayleigh quotient-based algorithm and efficiency enhancing integral formulas, we give the proofs that the bound for an infinite number of prime gaps is not more than 600, that there are bounded gaps for an arbitrary preassigned finite number of primes.We show that the constants in Maynard’s choice of upper bound for several results are optimal among bounds of the given form. Finally, using great combinatorial counting, we give Maynard’s proofthat prime k-tuples have positive relative density if counted appropriately.