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24 january 2018

Vicky Neale subtitles her popular and accessible book, *Closing the Gap*, as "the quest to understand prime numbers." But of course you can learn a lot about prime numbers from books like Neale's, without coming any closer to understanding them than when you first heard them defined.

Prime numbers are divisible only by themselves and 1. The number 12 is divisible by 1, 2, 3, 4, 6, and 12, making it highly useful for all sorts of purposes, but making it "composite," with additional factors beyond itself and 1. The numbers 11 and 13, on either side, are only divisible by themselves and 1; they are prime. That does not make them useless. 11 seems to be an ideal number for team sports. 13 is one too many for dinner, either the luckiest or unluckiest of numbers depending on your personal taste.

*Closing the Gap* is a book about pairs of prime numbers like 11 and 13, primes that differ by two. We have known since antiquity (thanks to Euclid) that there is no end to single prime numbers. Neale explains Euclid's proof of the infinity of prime numbers especially well, as I will demonstrate by re-explaining it not quite so well. Suppose there is a greatest prime number: every number greater is composite, with more factors than itself and 1.

Take all the prime numbers, multiply them together, and add 1. This (huge) sum cannot be divisible by any of the numbers you've just multiplied, because the smallest prime number is 2. So the sum is either prime (thus a prime greater than the greatest prime), or composite. If composite, it has a prime factor greater than the greatest prime – otherwise it would be divisible by one of the smaller primes. We reach a contradiction – and so, a greatest prime number cannot exist.

Cool! So it should also stand to reason that there are infinitely many prime numbers like 11 and 13, standing just two apart – "twin primes" in mathematical parlance. Indeed the "Twin Prime Conjecture," a very old idea, says just that. It seems pretty obvious. But it has so far been impossible to prove.

Like Neale writing her book in 2017, though, I have to keep checking the Internet as I write this review, because somebody might prove the Twin Prime Conjecture any minute now. As of today, it has been proven that there are infinitely many prime numbers that differ by no more than 246. (Though you should check again.) 246 seems like a bunch more than 2, I know. But just five years ago, it was first proved that there are infinitely many prime numbers that differ by no more than 70,000,000. Before that, nobody could prove anything of the sort.

*Closing the Gap* is thus the story of how researchers got from 70,000,000 to 246. Actually, though Neale is very gifted at explaining math, it is impossible for a layman like me, somebody who knows the difference between a mean and a median but got a C in college calculus, honestly to understand proofs involving "prime gap bounds." It seems hard enough for professional number theorists to understand them. Neale does her best to get me to see *what* has been proved, if not how or why. She uses the metaphor of a punch card. Let's say you have a punch card that fits over a number line, with little windows that show numbers along the line. If it has two windows next to each other, the only primes it can show are 2 and 3. If it has two windows that differ by two, it will show 3 and 5, 5 and 7, 11 and 13, and gazillions more; but we don't know if it will show an infinite number of twins.

If you have a punch card that is 600 spaces wide and contains 105 windows, cut just so (Neale shows how on page 121), you are sure of seeing at least two prime numbers no matter where you lay the card on an infinite number line. That was known in 2013, and presumably the operative "card" is now only 246 spaces wide – which might work as a literal card, and would be fun to play with, if you also had an infinite number line.

That is what we know, but you and I (unless you're a top number theorist, in which case you've clicked away in contempt already) have to take the result somewhat on faith. That we take it on faith is, however, in part thanks to its intuitive rightness. If there's an infinite number of numbers, there ought to be an infinite number of certain groupings of certain kinds of numbers. It would be bizarre if the Twin Prime Conjecture turned out to be *false*. Fortunately, as with the infinity of primes, the way to prove it true is to try to prove it false, and nobody's achieving that any time soon.

*Closing the Gap* is not a human-interest story. It conveys some facts about some of the "characters" involved in mathematics, but to my own delight it is not an "interview-driven" book. It puts technical knowledge first and does not subordinate it to intriguing personalities, which are naturally irrelevant to the math. Yitang Zhang, the mathematician who set the initial Gap at 70,000,000, would be a natural subject for a human-interest story. He was a teaching-faculty PhD, at times an adjunct and at times a Subway sandwich artist, laboring without much research support for fifteen years till he constructed his astounding proof. This is dramatic stuff, but Neale appropriately subordinates drama to objective achievement.

I have praised Neale's expository talents, but I will confess that I got only into Chapter 12, where she talks about sums of two square numbers, before irrevocably losing the thread of whatever the hell she was talking about. In my own praise, there are only 16 chapters, and half of them are "meta" descriptions of the progress in narrowing the 70,000,000 gap figure. Neale offers a very interesting account of a system called Polymath where multiple collaborators can work instantly together over the Internet and achieve far greater progress (on certain kinds of problems) than lone geniuses with chalkboards, *pace* Good Will Hunting. Neale makes the higher math seem like a fun, clean, lovely and convivial activity. *Closing the Gap* is an inspiring book to read, no matter how far you get in it before your brain springs a gasket.

Neale, Vicky. Closing the Gap: The quest to understand prime numbers. Oxford: Oxford University Press, 2017.