As a number theorist myself, people always ask me “Why are primes so interesting?” Next time it happens, rather than spending hours proving my case, I will just refer them to Vicky Neale’s book *Closing the Gap*. Written in an engaging and inclusive way, it makes a perfect read for beginners but it also picks up the pace fairly quickly, so even enthusiasts like myself are bound to enjoy it. In particular, it starts by defining prime numbers, and yet somehow in the space of 160 pages, Neale manages to take the readers on a journey to cutting edge research mathematics. But enough rambling, let me tell you a bit more about what you might find inside the book and what we liked about it.

## What is inside

I will try and provide a quick summary, without giving too much away. In the introduction Neale provides an interesting way of looking at tackling a research problem, by comparing the experience to rock climbing on uncharted terrain. Henceforth the chapters alternate between the mathematics behind one of the oldest unsolved problems in number theory, the *Twin Primes Conjecture*, and the story of what followed from the latest breakthrough.

In the first few chapters, we are introduced to the notion of a* prime*, and *twin primes*, which are a pair of primes that differ by $2$. As with many problems in number theory, the twin prime conjecture is easy to state but as you might guess from the name is still unsolved. Similarly, *Goldbach’s conjecture* and the existence of infinitely many primes of the form $2p+1$ where $p$ is a prime, called *Germain primes*, is not known.

And as Neale says “if primes are hard let’s try something else.” Thus in the chapters that follow she discusses problems similar, in some sense, to the ones before, that we know the answers to or that we can prove.

And let’s not forget Vicky’s mathematical pencil, which she uses to illustrate the distribution of primes.

Following, this brief introduction into the world of analytic number theory, she moves on to some more complicated results from *Ramsey Theory* and probabilistic number theory that were used in the most recent breakthrough. Moreover, she gives a very nice heuristic of why we expect the twin prime conjecture to be true.

And as this is a pop maths book, there are plenty of problems for you to puzzle over.

However, if the maths, in particular the $\log$’s become too much for you, or if you are already familiar with the topics covered, you can always turn to the other half of the book which deals with the recent history of progress in solving the* Twin Primes Conjecture. *And this story has it all! Started by a single mathematician working on his own, who managed to combine works that seemed unrelated to make a dent in the problem, improved by a global collaborative project (*the Polymath project*), and we are yet to see how this story will end. We do however get an intimate insight into the research world in recent years, and it is a lot more exciting and dynamic than one might expect!

## What we liked

In an increasingly connected world, the internet is changing the way research mathematics is done. Closing the gap provides a valuable insight into this new world of blogs and massive online collaborations by telling the story of some recent groundbreaking results in prime number theory. Neale uses real conversations with, and comments from, experts in the field to bring the reader into this world and introduce them to the people behind the proofs. We like how Neale has done a good job of keeping the book lively by skillfully weaving the main narrative with entertaining and relevant mathematics for the reader to ponder and puzzle over.