# Prime Suspects: The Anatomy of Integers and Permutations

###### Andrew Granville, Jennifer Granville, and Robert J. Lewis
Publisher:
Princeton University Press
Publication Date:
2019
Number of Pages:
232
Price:
22.95
ISBN:
9780691149158
Category:
General
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Benjamin Linowitz
, on
08/31/2019
]
In Prime Suspects Granville and Granville have performed something of a feat. They've written a graphic detective novel that is both interesting to read and yet simultaneously teaches its readers some deep mathematics. I read the book in a single sitting and absolutely loved it.  The story, beautifully illustrated by Robert J. Lewis, begins when police officers B. Green and T. Tao discover the body of Arnie Int, a murdered lieutenant in the Integer crime ring. Detective Jack Von Neumann is assigned the case, and enlists the help of Professor C. F. Gauss and his students Emmy Germain and Sergei Langer in order to try to investigate some surprising links between this homicide and that of Daisy Permutation, a ballet dancer described as "odd, outside of the Alternating Group, her family's business."

As the story progresses, Gauss and his students are led to investigate some striking commonalities between the ring $\mathbb Z$ of integers and the group $S_N$ of permutations on $N$ letters. The idea underlying these commonalities is that $1$ in $N$ permutations on $N$ letters is an $N$-cycle, and $1$ in $\log x$ of the integers around $x$ is prime. It's therefore reasonable to inquire about whether one can take a formula relating to some property of permutations that is given in terms of $N$ and substitute $\log x$ for $N$ to get a formula that holds for the integers. As an example, it's known that a typical permutation contains about $\log N$ cycles. If we substitute $\log x$ for $N$ then our analogy would seem to imply that a typical integer should have around $\log\log x$ distinct prime factors. And indeed, this is a famous result of Hardy and Ramanujan from 1919!

Needless to say, there's some terrific analytic number theory in this book. Deep results like the Erdos–Kac theorem are discussed, as are the connections between primes and irreducible polynomials defined over finite fields. At the end of the book appears a very friendly survey article by Andrew Granville which goes into the book's mathematical content in greater detail and provides references to the relevant literature.  In addition to the rich mathematics, the authors draw attention to several issues of cultural importance in mathematics. Through Germain's interactions with fellow student Langer we are given a brief glimpse into the role of women in mathematics and some of the obstacles they face. (On p. 55 there's a billboard listing the percentage of math professors in the US, UK, and Mexico that are women. The percentages are 32, 16 and 65 respectively.) The authors also touch on the relationship between a student and their adviser and, through the characters of Joe Ten-Dieck and Count Bourbaki, the influence of deep abstraction.

I have to admit, as a mathematician, I absolutely loved all of the avatars of famous mathematicians that appear in Prime Suspects. Indeed, you almost need a broom to sweep up all of the names being dropped! Moreover, the illustrations are full of fun easter eggs: the "Google A204189 in OEIS" graffiti in the Green and Tao scene, the mall with the $P=NP$ Computer Repairs and Clay's Problem Arcade, Jean-Pierre Serre Boulevard, etc.

It's very difficult to write a book on an advanced topic in mathematics that's accessible to math students and enthusiasts yet touches on contemporary research that is of interest to a broad swath of practicing mathematicians. Prime Suspects is such a book. And it's entertaining to boot. I recommend it in the strongest terms.

Benjamin Linowitz (benjamin.linowitz@oberlin.edu) is an Assistant Professor of Mathematics at Oberlin College. His research concerns the theory of arithmetic groups, a fascinating area lying at the intersection of algebraic number theory and differential geometry. He is also interested in the history of mathematics. His website can be found at http://www2.oberlin.edu/faculty/blinowit/.