# Prime Suspects

This book, or at least its main part, is a graphical novel or comic strip that involves a lot of references to mathematics. All these references, in both the plot and the characters, need some explanation which is given in a second smaller text part. In fact this book is only part of a broader project since the original version was written as a stage play with the intention to popularize some ideas from mathematics involving prime numbers and permutations. There is some music composed for it and there are plans for a film.

The idea of the plot is to expose that there are similarities between the "anatomy" of integers and of permutations. By anatomy one has to understand that primes are the building blocks for all (positive) integers just like cycles generate all permutations. The fiction is that these anatomies are literally impersonated by Arnie Int and Daisy Permutation (strange names!). Arnie and Daisy are murdered and professor Gauss, professor in mathematical forensics, is in charge of analysing the bodies. He has two assistants, Sergei Langer, delegated by detective Jack von Neumann as a liaison observer, and Emmy Germain, the true heroin of the story. Emmy observes that the anatomy of both bodies is very similar. Meanwhile detective Jack von Neumann is visiting clairvoyant Joe Ten-Dieck in the Pyrenees, and after his return can reveal the murderer. All these names obviously refer to mathematicians that lived in different centuries, but the characters are fictional and the whole story, the characters and the decors are set in the current time frame. To switch between different moments in the story, use is made of a narrating character count Nicholas Bourbaki who looks a lot like Orson Welles. Finally, there is also a reporter duo consisting of a sound man and camera man (Silent Bob and Barry Bell) that play the role of the lay reader by asking to explain what is going on when the specialists dive into mathematical jargon.

Most mathematicians involved in this plot are not difficult to guess, but the text at the end gives more explanation including a short description of what the real mathematicians did. There can be no doubt about Gauss and Von Neumann. The defenders of abstract mathematics are impersonated by of course Bourbaki, Serge Lang (Langer) and Alexander Grothendieck (Ten-Dieck). Emmy Germain is Sophie Germain but she got the first name of Emmy Noether. The reporter, Barry Bell is a mix of the mathematical biographers E.T. Bell and Barry Cipra. The cameraman Silent Bob is the tv and film character created and often played by Kevin Smith. Ben Green, Terry Tao, and Tamar Ziegler appear as cops. But besides these well known names, almost all the frames are populated by characters that have the face of historical or living mathematicians. The colourful graphics are very realistic (although not always anatomically perfect), drawn from many different perspectives, and with a lot of references in the detailed backgrounds.

Two frames have a QR code that bring you to relevant websites. Some examples of math references: cars have a licence plate that are prime numbers (but Von Neumann's is EDVAC1), a Klein bottle on a shelf, Warhol-like paintings, except that Marilyn Monroe is replaced by Gauss, a departure board in the airport announces Köningsberg, Hanoi, and Flatland as destinations, road signs are pointing to Langlands Highway or Jean-Pierre Serre Boulevard, and shops are called "P=NP computer repairs", "Perelman & Mum", or "Yang & Mills light points", one of them sells Noether's annuli. One can recognize portraits of mathematicians on the wall and posters with formulas or announcements, mathematical books are read that have recognizable titles, and many facts are of course explicitly mentioned: from Occam's razor to Grothendieck's *dessins d' enfant*. And there are fun phrases like Langer claiming that the victims are not related: "they are about as similar as apples and iphones". Steve Jobs might have appreciated that one.

Much of the mathematics is also explained in the appendix. One ingredient is the distribution of primes. Counting the number of primes less than a random number $x$, then on average about 1 in every $\log(x)$ numbers is prime and the $\log\log$ of the prime factors have a Poisson distribution. Something similar holds for permutations. Given the permutations of $N$ random items then the number of cycles has a Gaussian distribution with average $\log N$ and standard deviation $\sqrt{\log N}$, while the length of he cycles follow a Poisson distribution. A "calibration" $N \leftrightarrow \log x$, shows that the number of prime factors and the number of cycles are described by similar formulas. Also the log of the prime factors of a random number and the length of the cycles or a random permutation are described by similar formulas if he same calibration is used. Other similarities can be detected between integers and permutations. The same kind of "anatomy" exists in other structures as well. At the end of the novel, Emmy is giving an inaugural lecture where she adds Poly Nomial (the polynomials over finite fields) to the analogy. Such underlying universal similarity was already proposed by Grothendieck who was too advanced for his time and hence is a clairvoyant in the story. This similarity that is showing up again and again in different circumstances is a familiar beat, a melody that sticks and that you cannot get out of your mind. In the story it resonates in the brain of the two victims. It is the music of the primes (a reference to Marcus Du Sautoy's book), and it also refers to the title *Reverie in prime time signature* of a composition by Robert Schneider that he wrote at the ASI in Princeton in 2009 after attending a first reading of the script for *Prime suspects*. The score of his composition is also included in the text with some explanation, and it can be listened to when using one of the QR codes in the comic book.

The main characters behind the book are of course the authors. Andrew Granville is the mathematics professor whose topic is number theory. Jennifer Granville is film and theatre producer, writer, and director, and, in the case of this case, most importantly, Robert J. Lewis, the illustrator. This is really a great book with so many references to mathematical ideas, that you can read and reread and find every time new details hidden in the graphics. Also the mathematical ideas are not so trivial or common knowledge, although explained only briefly and without technicalities. The lay reader may find these extra explications too mathematical and be content by just savouring the comic book, but the interested mathematician can find more details in the references. Another interesting aspect is that the novel does not only explain mathematical concepts but that it also explains how mathematical research really works. We see Emmy struggle with the similarities that she knows must be there but that do not match perfectly. It is only after she moves to statistics that she detects the calibration that allows her to identify the victims as "identical twins", The story is a whirling detective story, the graphics are almost realistic (although sometimes the recognizability of the persons requires unnatural postures and especially hands are not always technically correct). In the text I saw the name of Grothendieck a number of times misspelled as Groethendieck, but these are only minor flaws that will not, and should not, harm the warm feel-good sensation it will induce in math lovers caused by the predominant mathematical geyser on which the comic book floats.

**Submitted by Adhemar Bultheel |

**23 / Jul / 2019