Vicky Neale has a degree in number theory and is now lecturer at the Balliol College, University of Oxford. She has a reputation to be an excellent communicator. This also shows in this marvellous booklet in which she gives a general introduction to the advances made in the period 2013-2014 in the quest for a solution of the twin prime conjecture. But she also explains how mathematicians think and collaborate.
The twin prime conjecture is claiming that there are infinitely many prime numbers whose difference is 2 like 3 and 5 or 11 and 13. It is easy to explain what prime numbers are, and it is even possible for anyone to understand Euclid's proof that there are infinitely many primes. The twin prime conjecture is however still one of the long standing open unsolved problems: easy to formulate and understand but hard to solve. Several attempts and generalizations were formulated. For example it can be claimed there are infinitely many primes whose difference is an even positive integer N. The twin prime conjecture corresponds to N = 2.
And then, in April 2013, Yitang Zhang could prove that the latter generalization holds for N equal to 70.000.000, a major breakthrough. Within a year N was reduced to 246. Neale presents the different steps that were obtained in this reduction almost month by month as a thrilling adventurous quest.
Scott Morrison and Terence Tao, two mathematical bloggers quickly used Zhang's approach to reduce the N to 42.342.946. Tim Gowers, another active blogger proposed a massive collaboration and a Polymath project was set up by Tao. This Polymath platform is a totally new way of collaboration between mathematicians that Gowers had proposed back in 2009. The blog is fully in the open and anyone who wants to take part can dump some guesses or partial ideas on the website. The results are published under the author name D.H.J. Polymath and the website shows who has collaborated in the discussion. Neale spends some pages to discuss this kind of collaboration and comments on its advantages and disadvantages. The project on the twin primes was numbered Polymanth8 and it turned out to be particularly successful. The problem that had been out for so long now progressed quickly because already in June 2013, N was down to 12.006. In July they reached 4.689.
But while in August 2013 Tao is announces to write up the paper with the Polymath8 result, another twist of plot occurs. James Maynard posted a paper on arXiv in November 2013 in which the bound N is brought down to 700. Independently Tao announced on his blog on exactly the same day that he used the same method to obtain a similar reduction. Using the new method the old Polymath8 was renamed as Polymath8a and a new Polymath8b project was started. This resulted in April 2014 in bringing the bound down to 246. The bound can even be 6, but that requires to assume that the Elliott–Halberstam conjecture (1968) holds, which is a claim about the distribution of primes in arithmetic progression.
But Neale in this booklet brings more than just the account of this thrilling quest to close the gap. She also succeeds in explaining parts of the proofs and she also tells about similar related problems from number theory. For example the Goldbach conjecture: "every even number greater than 2 is the sum of the squares of two primes", or its weak version: "every odd number greater than 5 is the sum of three primes", are two famous examples. The generation of Pythagorean triples is another well known example. But there are other, maybe less known ones like Szemerédi's theorem proved in 1976, which proves as a special case a conjecture by Erdős and Turán: "the prime numbers contain arbitrary long arithmetic progressions". The Waring problem: "every integer can be written as a sum of 9 cubes, or more generally, as a sum of s kth powers, (where s depends on k), which triggered Hardy and Littlewood to count the number of ways in which this is possible. They proved the Waring conjecture by showing that there is at least one way of doing that. Neale also explains admissible sets which were used in a theorem proved by Goldston, Pintz and Yıldırım which was essential in proving and improving Zhang's bound on N. And there is some introduction to the prime number theorem and the Riemann hypothesis.
Neale cleverly interlaces these diversions with the progress on the twin prime problem, which has the effect that some tension is built up and new developments pop up as a surprise. Some of the notions and terminology that popped up in the other problems turn out to be related or at least to be useful in the twin prime problem.
Neale realizes that she is writing for a general audience and carefully explains all her concepts. However, I can imagine that some of the mathematics, like for example the formulas for the asymptotics in the Hardy-Littlewoord theorem involving a triple sum, fractional powers, complex numbers, and gamma functions will be hard to swallow for some of her readers. On the other hand, many of her "proofs" rely on visual inspection of coloured tables, and she has witty ways of explaining some concepts. For example admissible sets are presented as punched cards, a strip with a sequence of holes at integer distances, and the idea is that when this is shifted along the line of equispaced integers, then at least one (or more) primes should be visible in the punched holes. Modulo arithmetic she explains using a hexagonal pencil with the numbers 1-6 printed on its sides at the top, then 7-12 next to it etc. If you put the 6 sides of the pencil next to each other, you get a table of numbers modulo 6, and the primes in this table show certain patterns. Some of the graphics are less functional, yet very nice. On page 6 where prime and composite numbers are explained, a prime number p is represented with p dots lying on a circle, while composite numbers are represented by groups of dots arranged in doublets, triangles, squares, etc. which gives a visually pleasing effect. Other graphics are referring to a pond with frogs, grasshoppers, ducks, reed and waterlily leaves. These may be less instructive, but they are still a nice interruption.
Vicky Neale has accomplished a great job, not only in bringing the mathematics and the mathematicians to a broad audience. We meet some of the great mathematicians of our time like Gowers and Tao, both winners of the Fields Medal. We are informed how mathematical progress works, how new ideas are born. This can be through novel communication channels such as the Polymath, but it can still be a loner who works on a completely different approach who comes up with a breakthrough. Sometimes we can gain from results slumbering in mathematical history, but often it relies on coincidences when someone connects two seemingly unrelated results. And when the time for an idea is ripe, then it happens that two mathematicians independently from each other come up with the same result simultaneously.