What can I say? I am a fan of Paulo Ribenboim, and I avidly look forward to reading his new books. His other works include The Little Book of Big Primes (1991), 13 Lectures on Fermat's Last Theorem (1995), The [New] Book of Prime Number Records (1998), and Fermat's Last Theorem for Amateurs (1999). His writing is almost always clear and warmly wry. Most importantly, he takes special pains to make certain that he includes the most recent results and an extensive bibliography.
So when My Numbers, My Friends came out, I was excited. And this book more than rewarded my anticipation!
Ribenboim is now retired, and treats us to a delightful survey of a few of his favorite numbers, including the ominous 65, which he relates, in a lively discussion in his final chapter Galimatias Arithmeticae, to Euler's numeri idonei. These "idoneal numbers" were convenient to him for proving that certain large numbers are prime. Let n be greater than or equal to 1. Consider the problem of representing numbers q in the form q = x2 + ny2, with x and y coprime positive integers. Some numbers won't have any such representation, some will have only one, and some will have many. Euler called n a numerus idoneus if the fact an odd number q that is uniquely representable in that form implies that q is either a prime or a prime power. Euler found 65 such numbers. It has been proven that there are only finitely many such numbers, and it has since been shown that there can be no more than 66 numeri idonei!
This book has a number of wonderful and wryly-written passages. Its coverage is vast: although much of the material in the lectures can be found in number theory textbooks, Ribenboim has integrated and consolidated so much related material from the literature that each lecture sparkles from its new treatment. As a collection of lectures, one expects to find some dated material. One doesn't! Indeed, the lectures have been brought right up to the publication date!
The topics covered include the Fibonacci numbers and Lucas sequences; continued fractions and representation of real numbers by means of Fibonacci numbers; updated prime number records (of course!); prime number production and cryptography; Euler's prime generating polynomial and its relationship to Gauss and the class number problem and binary quadratic forms. An article discusses consecutive powers: starting from the sequence of squares and cubes of integers in increasing order
4 8 9 16 25 27 36 49 64 81 100 ...
are there consecutive integers other than 8 and 9? Finitely many? How many? Are there three or more consecutive powers? From this question we move on to treatments of the Fermat residue and XU - YV = 1. The book also includes the famous 1093 lecture, which observes that 1093 is the smallest integer satisfying 2 p-1 º 1 (mod p2) and explains how this property has a fascinating relationship to the ABC Conjecture. There are discussions of Fermat's Last Theorem and the Wieferich congruence; transcendental numbers and the Gel'fand-Schneider theorem; and so much more. (References to wine, marriage and the Arctic Ocean will also keep the reader alert!)
However, and this is a big however, the book has all the flavor of having been thrown together far too quickly by the publisher. Instructors considering use of this book should be aware of a few potential pitfalls for students. The publisher's blurb states that these popular lectures "...are thoroughly accessible to everyone with an interest in numbers." Well, the lectures are sometimes written as densely with citations and results as Dickson's History of the Theory of Numbers, and the prerequisites go well beyond having an interest in numbers! This book builds significantly on advanced undergraduate and graduate level knowledge. At times, some obvious results weren't. Like many mathematics books, one reads this book by "rewriting" it, and of course one learns by rederiving the less obvious "obvious" results. But there are other cases where the fault was not the reader's.
I began spotting typos first in equations (a subscript here and there, a division-bar turning into a comma), but then errors in obvious elements of an enumerated set of integers (on page 176, 5 is listed as a power of 2 or 3). But it gets worse: Euler reportedly sent a letter to Goldbach in 1979 on page 281, and there is even a misspelling of the word 'said' as 'siad'. Some phrases are presented as sentences, even though they aren't. don't make sense without guessing. Citations sometimes fail to show up in the chapter's bibliography, and sometimes have disparities in dates between their being referenced in the text and their bibliographic entry proper. These errors, along with a few of Springer's convention of placing fractions having exponents or subscripts in both numerator and denominator inter-linea in reduced type, have proven to distract greatly from the contributions in the text (and also to lay far too much of a burden on the reader).
Ribenboim should not be faulted for these errors, as his eyesight has been failing. An editorial eye is what is missing here. I found myself comparing discussions with those in his prior books, especially his Book of Prime Number Records and his Fermat's Last Theorem for Amateurs. Both of these earlier books are more clearly and better edited than My Numbers, My Friends. Fie on the editor!
There is a great deal to enjoy in this book; I learned a good deal in going through the text. It is well worth the investment!
Marvin Schaefer (email@example.com) is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.
Fibonacci Numbers and the Arctic Ocean.- Representations of Real Numbers by Means of Fibonacci Numbers.- Prime Number Records.- Selling Primes.- Euler's Famous Polynomial and the Class Number of Imaginary Quadratic Fields.- Gauss and the Class Number Problem.- Consecutive Powers.- 1093.- Powerless Facing Powers.- The Classical Bernoulli Numbers.- Galimatias Arithmetica.- The Work of Kummer on Fermat's Last Theorem.- An Essay on Irrational Numbers.