Closing the Gap: The Quest to Understand Prime Numbers by Vicky Neale (PDF)

Description

In 2013, a little known mathematician in his late 50s stunned the mathematical community with a breakthrough on an age-old problem about prime numbers. Since then, there has been further dramatic progress on the problem, thanks to the efforts of a large-scale online collaborative effort of a type that would have been unthinkable in mathematics a couple of decades ago, and the insight and creativity of a young mathematician at the start of his career.Prime numbers have intrigued, inspired and infuriated mathematicians for millennia. Every school student studies prime numbers and can appreciate their beauty, and yet mathematicians’ difficulty with answering some seemingly simple questions about them reveals the depth and subtlety of prime numbers.Vicky Neale charts the recent progress towards proving the famous Twin Primes Conjecture, and the very different ways in which the breakthroughs have been made: a solo mathematician working in isolation and obscurity, and a large collaboration that is more public than any previous collaborative effort in mathematics and that reveals much about how mathematicians go about their work. Interleaved with this story are highlights from a significantly older tale, going back two thousand years and more, of mathematicians’ efforts to comprehend the beauty and unlock the mysteries of the prime numbers.

User’s Reviews

Editorial Reviews: Review “The way [Closing the Gap] explores mathematics and at the same time describes the work mathematicians do, is very interesting and it keeps the reader invested in the book. It is easy to read and precise. The book could be definitely recommended to mathematics students and teachers but also to younger people with an interest in higher-level mathematics.” — Panayiotis Vlamos , University of Athens, MAA”The book features a creative structure that lends itself well to the subject matter. A curious undergraduate mathematics major should enjoy this book and learn a great deal. For mathematicians who do not specialize in number theory but who are curious about the flurry of recent activity in the field, this book provides an excellent entry point.” — Stephan Ramon Garcia, Notices of the American Mathematics Society”If you are looking for an introduction to the world of Polymath; if you are looking for the story of the Twin Primes Conjecture; if you are looking to show you friends and family what your life as a mathematician is; if you would like a bit of asymptotic mathematics explained to you plainly; if you would like a summary of Waring’s problem; or if you just have a couple of hours and are looking for a nice diversion, then you have found it.” — Deborah Chun, London Mathematical Society”The book is clearly and enthusiastically written and beautifully presented.” — Owen Toller, The Mathematical Gazette”For myself, I learned a lot, even about subjects I thought I knew before… it is clear from every page in the book that Neale is superb teacher. In sum, I recommend this book highly to anyone interested in mathematics, young people and teachers but also to researchers.” — Mathematical Thinking and Learning”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. Neale manages to take the readers on a journey to cutting edge research mathematics.” — Chalkdust Magazine”Neale writes in an inviting style that draws readers into this challenging subject, convincing them that, with a little effort, they too can follow along. An enjoyable book and journey, complemented by a helpful reading list and index… Recommended.” — CHOICE”Closing the Gapis an excellent exposition of the study of prime numbers. Not only do we learn about the history of this area since the Greeks, but the book is the first aimed at a lay readership that provides insight into recent breakthroughs. Vicky Neale’s passion in the subject is contagious and I enjoyed how she weaves together the mathematics with background on how mathematicians now work, as well as her reflections on what it is like to be a mathematician. This book would be ideal for a curious sixth former wanting to peek ahead at what might lie around the corner if they are considering studying mathematics at a higher level.” — Alex Bellos, author of Alex’s Adventures in Numberland and Alex Through the Looking-Glass”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.” — MAA Reviews”Closing The Gap has gone straight into my top ten books to give to interested students… The book’s introduction starts with an extended analogy comparing mathematics to climbing [and] Neale sets herself up as this guide, and succeeds brilliantly.” — The Aperiodical”Closing the Gap is among the clearest popular accounts of maths I’ve read in a while. It’s about prime numbers, as the title suggests, but it’s also a master piece in the art of weaving. Apart from exploring the mathematics, the book gives an intimate description of the process of doing maths as experienced by those who do it every day, and an account of a particularly exciting, and recent, period when prime number theory made some great leaps forward. And it’s a look at a completely new way of doing mathematics: in large online collaborations that anyone can join.” — PLUS About the Author Vicky Neale, Whitehead Lecturer, Oxford UniversityVicky Neale is the Whitehead Lecturer at the Mathematical Institute and Balliol College, University of Oxford. A substantial part of her remit is public communication of mathematics. She has wide experience of working with students of all ages (school children to adults, via undergraduates), and of giving public lectures, and does various media work.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Closing the Gap is a highly readable popular math account of the recent progress made on the twin prime conjecture. For those unfamiliar, the twin prime conjecture is one of the famous elementary number theory questions that remains an outstanding problem. The twin prime conjecture is that there exist an infinite number of primes separated by 2. Such a simple statement has turned out to be extraordinarily difficult to prove. Closing the gap discusses the problem, basic ideas on number theory and the distribution of primes as well as the evolution of collaborative mathematics by greater use of online forums to aggregate specialist efforts. The book tells both a story as well as informs the interested reader about what doing math is really about. The book starts with setting the scene about how a mathematician working on his own, made a major step forward by proving there exist an infinite number of primes separated by a difference of less than 70 million. For the non mathematician, that seems like it is of no consequence to the settling of the twin prime conjecture, but the author quickly begins to shine light on the significance of the discovery. The author discusses how the online collaboration, polymath, soon cut down the bound on the difference between the primes as well as some basic ideas about how to think about sieve methods in number theory. The book is much less about math and much more about how to think about a mathematical problem and how real math is solved today. Certain star mathematicians are celebrated, Yitang Zhang, Terence Tao and James Maynard. The organization of the book is conducive to both keeping interest as well as making the reader think a little. In particular the author spends every other chapter following the evolution of the polymath project and the other half on discussing some particular math idea associated with the understanding the conjecture.The author embeds her love of math in the writing which helps absorb the reader in the book. There are certain mathematical ideas which are challenging to follow for the uninitiated, like the Hardy Littlewood circle method, but the reader gets a flavor of both how difficult and subtle a problem can be as well as how rewarding the subject can be when one makes a connection where none was seen before. Ironically the methods described in Closing the Gap wont be able to be used to prove the twin prime conjecture as the author highlights their are some insurmountable issues to the methodology, but the book is not about proving the conjecture so much as its about what doing math is all about. Enjoyable read, not too challenging while definitely interesting.

⭐I have to admit I haven’t done much research beforehand. I was looking for a book that would allow me to discover many secrets about prime numbers.After reading the book, however, the disappointment was great. The book has only 136 pages (11 blank pages, intro and index not included). The chapters are dates on which there were breakthroughs with regard to the twin prime conjecture, eg: On date X the gap was 70,000,000, and on date Y it was reduced to 30,000. There is a long discussion on who could narrow the gap and when. But no information is given on HOW this was handled, or the evidence itself (simply explained).Repeatedly it is said: “There’s lots of interesting stuff I could say here about this topic, but …” the writer leaves it to the reader to find out for themselves. Some things are explained in a childishly simple way, even on the basis of a few children’s drawings. Other concepts were too difficult for me to understand after reading once. A few chapters are interesting, e.g. the chapters about squaring primes. Admittedly, I learned some things. But above all, it is a disappointing feeling that is taking over, especially considering the cost of this book.

⭐To be truthful, I didn’t get it all first read through. I am about to start at the beginning again and am sure I will understand more next time (and the time after).And this is not meant to be a criticism, but I found this book (and many other demanding Kindle reads) superb for sending me to sleep. When I (inevitably) drop my Kindle, that’s the time to put it down on the bedside table. Then after a few thoughts about prime numbers, Il be asleep in seconds. I rather doubt that Vicky Neale has realised this hidden power of Primes.

⭐Well, I learned something, and in general it was as clearly written as possible for non-mathematicians. I’d reccommend it.4 not 5 stars because sometimes the illustrations which used “nature” like frogs and grass blades actually distracted from the idea. In some cases there is no need to pretend that a frog is jumping along prime numbers. It sounds “user-friendly” but adds nothing to the argument. We’re all adults here. Diagrams should be clear and not overly decorative, have a look at Tufte’s books for more info, for example “The Visual Display of Quantitative Information”

⭐Too many maths books seem to avoid showing actual mathematics, this one has plenty to get your teeth into. In order to maintain the flow of the mathematical story, the reader is often encouraged to prove some of the results: these are not too difficult but can be skipped without spoiling the arguments.What stands out in this book is the feeling that you’re gaining an insight into the pleasures, challenges and pitfalls of mathematical discovery. Very enjoyable.

⭐I enjoyed it. I needed to read it in chunks to give me time to take on the info as I’m not a mathematician

⭐Really enjoyed reading all of this book. I thought about some mathematics I knew and enjoyed getting my head around some results which were new to me. Additionally I was interested in the collaborative research description which is well beyond my experience!

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