Review of The Prime Number Conspiracy
The MIT Press, Cambridge, MA, 2018
336 pp. illus. 26 b/w. Paper, $19.99
This most excellent book is a collection of short articles from Quanta magazine. Under normal circumstances, a collection of articles written by a number of different authors gathered together is something best avoided, but this is a glorious exception. The topics although wide ranging have been selected to be both interesting and profound. The writers are excellent, the articles are very well researched and show a good understanding of difficult subjects, and the subjects are explained without recourse to any advanced mathematics, in fact without any mathematics at all. As a mathematician myself who understands the mathematics behind the articles, this almost impossible task has been achieved successfully by using analogy and very skilful prose.
The book is divided into seven sections, only the first of which is specifically about prime numbers. However, to be fair to the book, prime numbers do occasionally appear in other sections. This first section explores the distribution of twin primes (those that differ by two such as 101 and 103); estimating the number of primes as they get rarer along the number line; and the distribution of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 in individual primes. Subsequent sections cover other aspects of mathematics.
The second section concerns the occurrence of very specific numbers and mathematical structures in nature. Everyone knows about the Fibonacci sequence, but the structures examined here are lesser known. For example, a distribution called “universality” that looks random but isn’t. A distribution that looks Gaussian but isn’t (it’s called Tracy-Widom). Also, there’s a closer look at random processes and how particle physics and astronomy lead to the same periodic mathematical structures as those found in the solutions to simple algebraic equations.
The third section moves on to describe how some proofs are rescued from obscurity. Here the book branches into graph theory, statistics (again), and operational research (management science in the U.S.). Most of us realise that circles pack most closely in a kind of honeycomb pattern in two dimensions, but what about three and higher dimensions? As theoretical as this kind of problem sounds, such topological questions can provide solutions to fundamental problems in lattice theory, and these are applied to error correction in codes.
The next two shorter sections are on how computers can help to prove mathematical theorems and challenges on defining different infinities. The section on computers includes Russell’s paradox, whereby the set of all sets of elements that do not contain themselves causes a problem. Applying logic, if an element is in this set, then it isn’t. On the other hand, if an element is not in this set then by definition it has to be in there. As unresolvable as this seems, mathematicians have managed a work around by disallowing certain definitions of a set, and it is a good story. Infinity has always perplexed the thinking human, and to learn that that a whole hierarchy of infinities are now proved to exist still seems remarkable. The proof is technical, but here the prose is so well written it is almost as convincing. Finally, and a little bizarrely, is doing this kind of mathematics good for your mental well-being? This last section is philosophical and veers towards being a little homespun.
This book exudes class from every chapter. The readability is paramount, and each author achieves this readability not only by excellent prose style but by writing in detail about the people in a sympathetic way. The biographical detail is enough for you to get to know the person behind the mathematics. It is just worth mentioning that, as excellent as this book is, there are two minor problems. First, the Fields medal keeps being introduced as if for the first time. Secondly, there are portions where the writing style goes into a direct interview like a play, which is lazy writing. Both of these are pitfalls more prevalent in multi-author works. They could have been easily put right by careful editing. Apart from this quite minor point, each chapter reads very well, less like an essay and more like a short story. This book certainly deserves a very wide readership.