# Prime Numbers, Friends Who Give Problems: A Trialogue with Papa Paulo

Paulo Ribenboim is a number theorist, born in Brazil in 1928, who is living in Canada since 1962 where he was professor at Queen's University. In the tradition of the Socratic dialogues, he wrote this trialogue on prime numbers in which he is Papa Paulo and his opponents are Eric and Paulo. These two are interested in prime numbers. In fact, it is Eric who starts asking elementary questions about primes to which Papa Paulo answers. Soon Eric is joined by his friend Paulo (the other Paulo), and Papa Paulo is renamed to be P.P. (for obvious reasons he is not really happy with that alias). I believe Ribenboim wants to picture Eric and Paulo as young adults, and this is how I first imagined them, also since P.P. is addressing them as such, but then they seem to be very knowledgeable about many things (for example they point out to P.P. that Édouard Lucas was French or Eric who is said "to have travelled a lot"), things you would not expect from teenagers asking elementary questions about prime numbers. Whatever they are, it is just a minor glitch in the story, which does not affect the mathematics.

The discussion thus starts at a very elementary level but after a while it gradually turns into a course on prime numbers with formulas, computations, theorems and proofs. There are some intermissions in italic like *`Eric paused for a while, then continued'*. Here is another one: P.P. tells that Fermat after his death meets Saint Peter who has to decide on whether he should be sent to Heaven, Hell, or Purgatory. Fermat is confronted with his little lie about the short proof that he had for his last theorem but that the margin was too small to contain it. In that chapter P.P. is discussing the primality of Fermat numbers and states at the end that it is not known whether there are infinitely many Fermat numbers that are prime or that are composite. Then that chapter ends with the funny remark: `*The effect of this strong statement of ignorance caused this reaction on Paulo and Eric: Poor Fermat, he may stay in purgatory forever.*'

The latter illustrates that the conversation that has mostly a serious mathematical aspect, also has instances with funny components. Besides these few italic parentheses and some notes at the end of the chapters in which some biographical notes are added about a person that was mentioned (Euclid, Euler, Mersenne, Legendre, Fermat, and many many more), the whole book is just reproducing the conversation among the three protagonists. There is another bit of an unrealistic aspect to this trialogue when it comes to all the computations and formulas or formulations of theorems with their proofs. The latter formulation include the titles `Theorem' and `Proof' in bold and end with an q.e.d. message. This is something you only find in a printed mathematics book, not in a conversation, unless the discussion is taking place while the characters are writing down what they are saying as it is printed. This is indeed how we should read it because at some point P.P. *says*: `You are sharp-eyed, but what I *wrote* is correct.' (my emphasis). So he *wrote* it, not *said* it. Although P.P. is in fact to be identified with the author (Ribenboim is the meta-P.P.), he basically only reproduces the conversation and does not tell us much of the meta-story about the who, how, what, and where of the actors outside what is in the conversation. So there is only a very thin sketch of their personality, and only few circumstantial remarks in the conversation go besides the mathematical discussion. Ribenboim is just following the literary genre of the Platonic dialogues seasoned with contemporaneity and humour.

Although the reading is light, the book is not easy for a truly unskilled reader since, as the book advances, the mathematics get more and more involved. In the beginning it is about the Euclidean algorithm, gcd, lcm, modular arithmetic, the Wilson theorem, Fermat numbers, and Mersenne primes, up to primality testing and public key encoding. But when it comes to the prime number theorem, it requires real numbers, the log and exp functions and the logarithmic integral and for the formulation of the Riemann hypothesis, complex numbers and complex functions, series, analytic continuation and much more advanced mathematics need to be introduced. Nevertheless, the `technical stuff' is left out as much as possible. At some point, one of the intermissions read: *Papa Paulo was visibly happy with the presentation of the important theorem of Dirichlet on primes in arithmetic progressions. He was particularly elated to have been able to hide all the technical innovation needed to prove the theorem in its general form...'*

Towards the end of the book many more curious facts and conjectures about prime numbers are formulated (twin primes and the likes, conjectures by Goldbach, Sophie Germain, Bunyakovskii, Schinzel and Sierpinski, and many others). There are even conjectures by Papa Paulo and by Eric.

Paulo Ribenboim has written some dozen books almost all published by Springer. This one is published by World Scientific, so I do not know how much is fiction and how much is truth, but the last chapter is about publishing the notes of the trialogue. P.P.'s usual (fictional) publisher Marcel Spank at Gold Springs Publishing Company New York does not like the original title *Prime Experiments Explained to Boys and Girls* and proposes *The Story of Two Boys in Love with Prime Numbers* (from this it should be understood that Eric and Paulo are indeed boys and not adults). Eventually Spank turns down the manuscript. This at the time of P.P. writing this chapter it is still uncertain whether the notes will be published or not. It is also in this chapter that P.P. gives his unconventional idea about why there are so few women in mathematics: He, being a man, gets his best ideas while shaving, and women don't shave, hence....

As a conclusion, I liked the book and at some stages it is absolutely funny. The reader should however be prepared to swallow all the mathematics, the theorems, the proofs, the formulas and all the computations. As long as only integers are involved, in principle anybody motivated enough can understand what is going on. When it becomes more involved around the formulation of the prime number theorem, it may become a bit more difficult to hang on, but then it becomes interesting again when all these mysterious properties about prime numbers are conjectured. I can imagine that it will get smart young people interested in starting a career involving number theory.

**Submitted by Adhemar Bultheel |

**2 / Dec / 2016