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Index
Title page
Copyright
Contents
Chapter 1 Introduction
Chapter 2 What is a prime?
Patterns in the primes
What else might we ask ourselves?
Gaps between primes
Chapter 3 May 2013
Chapter 4 It’s easy to ask hard questions
Goldbach’s Conjecture
Germain primes
So are the primes impossibly hard to understand?
Chapter 5 May 2013
Chapter 6 Making hard problems easier
Almost primes
The weak Goldbach Conjecture
The Prime Triples problem
Szemerédi’s Theorem
Prime 5APs
More about evenly spaced primes
Chapter 7 June 2013
Punch-cards and admissible sets
Goldston, Pintz and Yildirim
Improving on Zhang’s bound
Polymath
More Polymath projects
Polymath5
Polymath8
Chapter 8 How many primes are there?
How many twin primes?
Chapter 9 July 2013
The parity problem
Chapter 10 What’s so mathematical about my mathematical pencil?
Are there infinitely many primes that are one less than a multiple of 6? Are there infinitely many primes that are one more than a multiple of 6?
Different sorts of pencil
Are there more primes that are one more than a multiple of 6 or more that are one less than a multiple of 6?
Chapter 11 August 2013
Chapter 12 If primes are hard, let’s try something else
Which numbers are sums of two squares?
Sums of two squares (continued)
Multiplying sums of two squares
Interlude on prime factorisations
Which (prime) numbers are sums of two squares?
Sums of two squares: the conclusion
Chapter 13 November 2013
Chapter 14 Generalise …
Sums of more squares
Modular arithmetic
Back to sums of three squares
Sums of four squares
A different sort of generalisation
The Hardy–Littlewood circle method
The asymptotic formula
The singular series
Ongoing work on Waring’s problem
More on the Hardy–Littlewood circle method
Chapter 15 April 2014
Chapter 16 Where next?
Large gaps between primes
Is Polymath the future?
The future
Further reading
General mathematics books and websites
Number theory books and websites
Polymath
Bounded gaps between primes—the start and Yitang Zhang
Bounded gaps between primes—the Polymath project and James Maynard
More on prime numbers
Index
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