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Closing the Gap: The Quest to Understand Prime Numbers
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 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.
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 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.
172 pages, Hardcover
Published December 12, 2017
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Displaying 1 - 9 of 9 reviews
October 12, 2017
Every now and then a working scientist will write a superb popular science book, but it's significantly rarer that mathematicians stray beyond recreational maths without becoming impenetrable, so I was cheering as I read the first few chapters of Vicky Neale's Closing the Gap about the attempt to prove the 'twin primes conjecture' that infinitely many pairs of prime numbers just two apart.
I'd say those first few chapters are far and above the best example I've seen of a mathematician getting across the essence of pure maths and why it appeals to them. Unfortunately, though, from then on the book gets bogged down in the problem that almost always arises, that what delights and fascinates mathematicians tends to raise a big 'So what?' in the outside world.
Neale interlaces attempts getting closer and closer to the conjecture, working down from a proof of primes several millions apart to under 600, adding in other, related mathematical work, for example on building numbers from squares and combinations of primes, but increasingly it's a frustrating read, partially due to necessary over-simplification. Time and again we're told about something, but effectively that it's too complicated for us to understand (or we'll come back to it in a later chapter), and this doesn't help make the subject approachable. I understand that a particular mathematical technique may be too complicated to grasp, but if so, I'm not sure there's any point telling us about it.
Part of the trouble is, most of us can only really get excited about maths if it has an application - and very little of what's described here does as yet. I'm not saying that pure mathematics is a waste of time. Not at all. Like all pure research, you never know when it will prove valuable. Obscure sounding maths such as symmetry groups, imaginary numbers and n-dimensional space have all proved extremely valuable to physics. It's just that while the topic remain abstract, it can be difficult to work up much enthusiasm for it.
At the beginning of the book, Neale draws a parallel with rock climbing, and that we are to the mathematicians scaling the heights like someone enjoying a stroll below and admiring their skill. And, in a way, this analogy works too well. We can certainly be impressed by that ability - but a lot of us also see rock climbing as a waste of time and consider it as interesting if you aren't actually doing it as watching paint dry.
It's not impossible to make obscure mathematics interesting - Simon Singh proved this with Fermat's Last Theorem. But that was achieved with writing skill by spending most of the book away from the obscure aspects. I'm beginning to suspect that making high level mathematics approachable is even more difficult than doing that maths in the first place.
I'd say those first few chapters are far and above the best example I've seen of a mathematician getting across the essence of pure maths and why it appeals to them. Unfortunately, though, from then on the book gets bogged down in the problem that almost always arises, that what delights and fascinates mathematicians tends to raise a big 'So what?' in the outside world.
Neale interlaces attempts getting closer and closer to the conjecture, working down from a proof of primes several millions apart to under 600, adding in other, related mathematical work, for example on building numbers from squares and combinations of primes, but increasingly it's a frustrating read, partially due to necessary over-simplification. Time and again we're told about something, but effectively that it's too complicated for us to understand (or we'll come back to it in a later chapter), and this doesn't help make the subject approachable. I understand that a particular mathematical technique may be too complicated to grasp, but if so, I'm not sure there's any point telling us about it.
Part of the trouble is, most of us can only really get excited about maths if it has an application - and very little of what's described here does as yet. I'm not saying that pure mathematics is a waste of time. Not at all. Like all pure research, you never know when it will prove valuable. Obscure sounding maths such as symmetry groups, imaginary numbers and n-dimensional space have all proved extremely valuable to physics. It's just that while the topic remain abstract, it can be difficult to work up much enthusiasm for it.
At the beginning of the book, Neale draws a parallel with rock climbing, and that we are to the mathematicians scaling the heights like someone enjoying a stroll below and admiring their skill. And, in a way, this analogy works too well. We can certainly be impressed by that ability - but a lot of us also see rock climbing as a waste of time and consider it as interesting if you aren't actually doing it as watching paint dry.
It's not impossible to make obscure mathematics interesting - Simon Singh proved this with Fermat's Last Theorem. But that was achieved with writing skill by spending most of the book away from the obscure aspects. I'm beginning to suspect that making high level mathematics approachable is even more difficult than doing that maths in the first place.
April 3, 2020
This book unlocked a new world for me. My mind is a little blown. I definitely want to dig some more on the side and circle back to some of the concepts that were introduced at the end once I have a little more knowledge of things like complex numbers under my belt. 10/10!
February 9, 2025
(I think 3.5 stars would be my preferred rating, if half-stars were allowed).
I read this book as someone with a PhD in (pure) mathematics, so I was not the intended audience for this book (i.e. a general non-mathematical audience), and it's therefore a bit hard for me to review it.
I accept that it's a very difficult job, trying to describe and explain pure mathematics ideas and research to a general audience - it's something I've really struggled with over the years. I think the author did quite a good job of this, although, again, it's hard for me to be the judge of that.
I found the book clearly written, error-free, and fairly engaging, although it did take me over a year from starting the book to finally finish, so I did tire of it at times. I particularly enjoyed the explanations of elementary proofs (i.e. the proofs, and parts of proofs, that the author did give in detail) - I hope that a general audience was able to follow along.
Nevertheless, the book, and especially its conclusion, is kind of... underwhelming. Both the unfinished status of the ultimate problem in the book (the twin primes conjecture), but also because the actual detail of the mathematics that has got us to the current partial result, is nevertheless basically impenetrable for all but the total experts. But hey, that's generally the nature of pure mathematics. Maybe I'm just not cut out for it...
I saw Vicky Neale speak once, I think it was in 2017 at the Young Researchers in Mathematics conference at the University of Kent in Canterbury. The same year that this book was published. I think I remember being quite impressed by her, and the talk that she gave. I was very sad to learn that she died in 2023, at only 39 years old.
Being familiar with Vicky's voice meant that, in my head, she was narrating the book in my head as I read along. I generally quite like it when that happens, including in this case.
Thank you Vicky.
I read this book as someone with a PhD in (pure) mathematics, so I was not the intended audience for this book (i.e. a general non-mathematical audience), and it's therefore a bit hard for me to review it.
I accept that it's a very difficult job, trying to describe and explain pure mathematics ideas and research to a general audience - it's something I've really struggled with over the years. I think the author did quite a good job of this, although, again, it's hard for me to be the judge of that.
I found the book clearly written, error-free, and fairly engaging, although it did take me over a year from starting the book to finally finish, so I did tire of it at times. I particularly enjoyed the explanations of elementary proofs (i.e. the proofs, and parts of proofs, that the author did give in detail) - I hope that a general audience was able to follow along.
Nevertheless, the book, and especially its conclusion, is kind of... underwhelming. Both the unfinished status of the ultimate problem in the book (the twin primes conjecture), but also because the actual detail of the mathematics that has got us to the current partial result, is nevertheless basically impenetrable for all but the total experts. But hey, that's generally the nature of pure mathematics. Maybe I'm just not cut out for it...
I saw Vicky Neale speak once, I think it was in 2017 at the Young Researchers in Mathematics conference at the University of Kent in Canterbury. The same year that this book was published. I think I remember being quite impressed by her, and the talk that she gave. I was very sad to learn that she died in 2023, at only 39 years old.
Being familiar with Vicky's voice meant that, in my head, she was narrating the book in my head as I read along. I generally quite like it when that happens, including in this case.
Thank you Vicky.
February 18, 2021
nah no me gusto. era un poco rambly, y no parecia tan structured y menciona todos estos methods y proofs por nombre y cada single vez clarifica que oops que cosas no lo voy a explicar aqui.
dicho eso me habia gustado su vdeo que tenia en el oxford youtube channel como una presentacion para las niñas de y11 ( https://www.youtube.com/watch?v=VSw5R... ). era claro y aunque de nuevo muy basico, al menos te introducia al subject y no se sentia como una chore como fue este libro.
presenta el conjecture hablando de acceptable punchcards y asi introduce el tema. de aqui, habla del discovery de zhang que fue el primero en dar un limite a este tipo de conjecture diciendo que habia un infinite number of primes that were at most 70000000 numbers apart. habla muy superficially del sieve method y luego introduce el trabaja de terry tao, maynard y polymathy y los muchos incremental sucesses para bajar el limite hasta 246 y que si se acepto un eliot no se que conjecture el limite baja hasta 6.
habla tambien del goldbarch conjecture y introduce los solved problems que son bastante simple. esta parte es probablemente la unica part que me gusto un poco donde habla de what numbers can be expressed as the sum of two squares, y de 3 squares (todos menos los multiple de 7, esta parte lo explica to simple con modular arithmetic y me gusto un poquitin) hasta llegar a 4 donde todos los numberos pueden ser expressed como sum de four squares.
dicho eso me habia gustado su vdeo que tenia en el oxford youtube channel como una presentacion para las niñas de y11 ( https://www.youtube.com/watch?v=VSw5R... ). era claro y aunque de nuevo muy basico, al menos te introducia al subject y no se sentia como una chore como fue este libro.
presenta el conjecture hablando de acceptable punchcards y asi introduce el tema. de aqui, habla del discovery de zhang que fue el primero en dar un limite a este tipo de conjecture diciendo que habia un infinite number of primes that were at most 70000000 numbers apart. habla muy superficially del sieve method y luego introduce el trabaja de terry tao, maynard y polymathy y los muchos incremental sucesses para bajar el limite hasta 246 y que si se acepto un eliot no se que conjecture el limite baja hasta 6.
habla tambien del goldbarch conjecture y introduce los solved problems que son bastante simple. esta parte es probablemente la unica part que me gusto un poco donde habla de what numbers can be expressed as the sum of two squares, y de 3 squares (todos menos los multiple de 7, esta parte lo explica to simple con modular arithmetic y me gusto un poquitin) hasta llegar a 4 donde todos los numberos pueden ser expressed como sum de four squares.
October 23, 2018
I was in Vicky Neale's audience when she mentioned this book so I bought it.
I loved this and read it in no time - I'm a poor data point for her though as I got a DPhil in Combinatorial Number Theory in 1978. I understood it all immediately since I had studied it all decades ago (not the newer stuff surrounding Zhang, of course, but I knew just what she was on about). Lovely. Beautiful stuff.
Her mission, though, is to bring this to everyone and anyone, so it's one of those carefully considered "science books for the masses". She has some credentials in public engagement stuff, and I suspect does a good job of it. I did feel from time to time that she was asking a lot of the under-informed, and now and then the personal style grated, but that's just me.
I'd really like to talk to someone who doesn't know any undergrad maths who has read it to gauge her success. I hope it's good.
I'm going to use this is a maths workshop for VI formers.
I loved this and read it in no time - I'm a poor data point for her though as I got a DPhil in Combinatorial Number Theory in 1978. I understood it all immediately since I had studied it all decades ago (not the newer stuff surrounding Zhang, of course, but I knew just what she was on about). Lovely. Beautiful stuff.
Her mission, though, is to bring this to everyone and anyone, so it's one of those carefully considered "science books for the masses". She has some credentials in public engagement stuff, and I suspect does a good job of it. I did feel from time to time that she was asking a lot of the under-informed, and now and then the personal style grated, but that's just me.
I'd really like to talk to someone who doesn't know any undergrad maths who has read it to gauge her success. I hope it's good.
I'm going to use this is a maths workshop for VI formers.
July 26, 2025
A beautiful book about mathematics and mathematicians. It uncovers the magic of pure mathematics and the questions that people find. It helped me remember that feeling of solving puzzles I had when I was a child.
It makes the famous figures closer and puts you in the middle of action. It shows that mathematics is not an individual work of young geniuses, there is collaboration, people can make discovery at different age.
It also shows how mathematics is connected and how proofs are based on other proofs.
Book tries to give you tasks you can solve but then around page 100 it gives you really complex task and it is even hard to follow the proof.
It makes the famous figures closer and puts you in the middle of action. It shows that mathematics is not an individual work of young geniuses, there is collaboration, people can make discovery at different age.
It also shows how mathematics is connected and how proofs are based on other proofs.
Book tries to give you tasks you can solve but then around page 100 it gives you really complex task and it is even hard to follow the proof.
July 14, 2021
Good, will explained and got the excitement across.
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
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
June 22, 2024
Some parts are pretty poorly written. The beginning was good. Don’t think I’m too interested in number theory.
October 10, 2025
Closing the Gap is an excellent narrative on the history of and recent progress on the Twin Primes Conjecture, facilitated by large-scale, web-based collaboration inspired by the breakthrough work of Yitang Zhang in 2013. Prior to reading this book, I was familiar with the statement of the conjecture but not with any of the strategies undertaken to prove it.
The book is divided into sixteen chapters that switch between describing the conjecture and approaches to "closing the gap" between twin primes (as well as approaches to solve related problems) and narrating monthly progress attained through the Polymath Project, inspired by success of the parallel Polymath Project on the Density Hales Jewett problem that concluded some years before and initiated by Terence Tao.
It is no small task to write exposition on theoretical mathematics problems, and no more smaller a task to do so for a general audience. Coming from a mathematics background, I was already familiar with some of the concepts that Neale defined to explain mathematical progress on number theory problems but, even so, found the language to be helpful even for specialized audiences and non-patronizing to non-specialized audiences.
Closing the Gap is a short, "easy" read so far as theoretical mathematics go, though certainly requiring more focus than a typical popular mathematics book. For interested lay readers, I regard it as a perfect text for learning about the sorts of problems that interest pure mathematicians and how problems in pure mathematics are approached both individually and collaboratively.
The book is divided into sixteen chapters that switch between describing the conjecture and approaches to "closing the gap" between twin primes (as well as approaches to solve related problems) and narrating monthly progress attained through the Polymath Project, inspired by success of the parallel Polymath Project on the Density Hales Jewett problem that concluded some years before and initiated by Terence Tao.
It is no small task to write exposition on theoretical mathematics problems, and no more smaller a task to do so for a general audience. Coming from a mathematics background, I was already familiar with some of the concepts that Neale defined to explain mathematical progress on number theory problems but, even so, found the language to be helpful even for specialized audiences and non-patronizing to non-specialized audiences.
Closing the Gap is a short, "easy" read so far as theoretical mathematics go, though certainly requiring more focus than a typical popular mathematics book. For interested lay readers, I regard it as a perfect text for learning about the sorts of problems that interest pure mathematicians and how problems in pure mathematics are approached both individually and collaboratively.
Displaying 1 - 9 of 9 reviews