Let’s begin with some background. One of the more famous currently unsolved problems in number theory is the *twin primes conjecture*, which asserts that there are infinitely many pairs of consecutive odd numbers (such as 3 and 5, or 5 and 7) that are both prime. According to an article by Dana Mackenzie in *What’s Happening in the Mathematical Sciences, Volume 10*, it is unclear when, and by whom, this conjecture was first proposed, but its roots go back at least as far as 1849, when Alphonse de Polignac published the more general conjecture that for *every* positive even integer \(k\), there are infinitely many pairs of primes differing by exactly \(k\). Not only has the special case \(k=2\) (which is, of course, the twin primes conjecture) proved to be exceptionally difficult; until recently there was no proof that there exists *even* *one* \(k\)for which this result is true (the “bounded gap” problem).

In May 2003, however, Yitang Zhang electrified the number theory community by proving that there is in fact at least one such \(k\), and that indeed \(k\leq 70,000,000\). Then, in the year or so following Zhang’s announcement, there was a frenzy of activity in the area, and by April 2004 the bound for \(k\) had been reduced from \(70,000,000\) to \(246\).

What makes this story doubly interesting from a human interest (as well as mathematical) standpoint is that Zhang was not, at the time of his result, a professor at a prestigious research university; in fact, he was a lecturer at the University of New Hampshire, having experienced some difficulty getting a job, likely as a consequence of a bad breakup Zhang had with his doctoral advisor. He was also in his mid-50s. (Take *that*, G.H. Hardy.) His story makes interesting reading: indeed, the *New Yorker* magazine published an article about him in its February 2, 2015 issue.

In this excellent book, author Vicky Neale attempts, with considerable success, to make the story of the bounded gap problem accessible to laypeople with little formal training in mathematics. In the process, she also touches upon a lot of other topics. These include other topics in substantive mathematics (examples: the prime number theorem, Goldbach’s conjecture, sums of squares, Waring’s problem, the Hardy-Littlewood circle method, quaternions, Fermat’s Last Theorem and unique factorization of integers) as well as issues about mathematical culture. She does a superb job, for example, of explaining the nature of mathematical research, and ways of communicating mathematics. The book discusses the concept of peer-review in journals, the ArXiv, and (quite extensively, since it is very relevant to the bounded gap problem) the Polymath project, along with practical implications for this, such as attribution of credit. (I am often surprised by how little some of my math-major students know about things like math journals and the publication process.)

The structure of the book is interesting. The title of every odd-numbered chapter is a date (e.g., May 2003) and that chapter covers developments in the bounded gap problem during that month. The even-numbered chapters develop the necessary mathematics, starting with the definition of a prime number and the well-known proof (from Euclid) that there are infinitely many of them. The last chapter of the book is entitled “Where Next?” and poses some questions and speculations about the future.

Following this, there is a good bibliography, listing books, published articles, and internet sites (with URLs correct at least as of January 2017). The *New Yorker* article referred to previously, for example, appears in this bibliography. Even better, the bibliography is annotated; Neale not only lists the sources but describes what is found there.

Writing a book about mathematics for a lay audience can be a surprisingly difficult undertaking, because there are competing objectives to be balanced. If you write in too elementary a manner, the mathematics becomes so watered-down as to be useless; write too precisely, however, and you confuse your target audience. Neale manages to thread this moving needle nicely. Her prose is clear but not patronizing, precise but accessible. The result is a very enjoyable book that can be read with profit not only by laypeople but also by mathematics students and the people who teach them.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.