Power Play

The first two sentences of the preface of Ed Barbeau's "Power Play" give a strong hint of what kind of experience lies ahead for the reader. As a youth, Barbeau was introduced by his grandfather to the amusing properties of multiples of 142857, and that "... was the beginning of a fascination with numbers, which I continue to nurture on public transportation and in queues." As one who often factors house numbers (e.g.) into primes while jogging, I looked forward to finding much to enjoy in this book, and was not disappointed.

Barbeau, who edits the "Fallacies, Flaws, and Flimflam" column of The College Mathematics Journal, gives clear, simple descriptions, without wasting words. As the title of the book suggests, the focus is on material involving powers of integers. The first part of each chapter introduces ideas mainly through numerical illustration of patterns or properties. When the time comes for a precise formulation or an interesting generalization, a number in boldface invites the reader to pursue the topic further in later components of the chapter, usually presented as Exercises on the Notes (hints, suggestions, clarification of what needs to be proven, etc.), Notes (solutions, or references to solutions), Additional Exercises, and Solutions (or references) for these. This somewhat cumbersome arrangement results from an attempt to make the book's material accessible to readers at various levels, from secondary students (who may want or need to stop with the numerical examples) to college teachers (perhaps tracking down student project material). Navigating one's way through the various elaborations of a particular problem can thus be something of a chore, but not one of insurmountable proportions.

Chapter 1 (Odd Integers and Squares) begins with the familiar fact, presented both numerically and geometrically, that the first n odd natural numbers add up to n². Interpreting the restriction to odd integers as a sieving process, the author presents modifications of the process producing cubes and fourth powers as sums, and we're only up to page 2. By page 6 we've progressed to the fact that adding the sum of the first n fifth powers to the sum of the first n seventh powers gives twice the fourth power of the sum of the first n natural numbers (a 1914 Monthly problem).

Chapter 2 (Pythagorean Triples and Their Relatives) takes off from recipes for generating infinitely many Pythagorean triples in which the two smallest numbers, or the two largest, differ by 1; proceeds through some generalizations of the Pythagorean equation, noting in particular a fourth-order magic square of squares; and points out that Ramanujan's taxicab number 1729 = 12³ + 1 is one of infinitely many "near misses" (sum of two cubes being 1 more or 1 less than another cube) that Ramanujan had found. Here and in other chapters, seeming curiosities turn out to be instances of general patterns, and we should be grateful to Barbeau for tracking down so many of them for us.

Chapter 3 (Sequences) uses a Chapter 2 idea to build a linked chain of Pythagorean triples

He then observes that each middle number is a multiple of the preceding one, and notes that the multipliers follow a pattern similar to that in Euclid's proof of the infinitude of primes. A section on second-order recursions concentrates on combinations of squares and cubes in the prototypical Fibonacci example, while another section intertwines powers of 2, entries in Pascal's triangle, and the Catalan numbers.

Chapter 4 (Pell's Equation) [no mention of the misattribution] presents a routine for generating a solution of x² - 61y² = 1, notes an application of Pell's equation in deciding when the sum of the first n natural numbers will be a perfect square, and relates second-order recursions to Pell's or Pell-like equations. A brief section on such equations of higher degree is expanded in an appendix placed at the end of the book.

Chapter 5 (Equal Sums of Equal Powers) considers the problem of partitioning a set of natural numbers into two or more subsets so that the sums of the numbers in the subsets are the same, or so that the sums of the squares of the numbers in the subsets are the same, or both, as in {1, 4, 6, 7} and {2, 3, 5, 8}. The process is extended by imposing more conditions and considering higher powers, with a fairly impressive generalization stated in the Notes. Toward the end of the chapter, a nice connection is made with the magic square of Dürer's "Melancholia."

Chapter 6 (Digits and Sums of Powers) notes that 153 = 1³ + 5³ + 3³, and that 370, 371, and 407 are the only other such numbers. All numbers having a similar property for exponents 4 through 9 are collected in a very short list, and the absence of the exponent 2 is explained (although numbers written in any odd base can equal the sum of the squares of their digits). There is a natural transition to standard material on integers expressible as sums of two or three squares, and a statement of Lagrange's result on sums of four squares, followed by a discussion of Waring's problem. The chapter ends with some equations whose solutions contain all ten digits exactly once, or all nine non-zero digits exactly once, and other such oddities.

Chapter 7 (Interesting Sets) presents a list of sets having amusing (but presumably not very deep or significant) properties, such as {1, 22, 41, 58}, in which any 3-element subset adds up to a perfect square. It then concludes with products of integers differing from a square by 1, with the Fibonacci sequence as prototype again.

"Power Play" is a fascinating book, densely packed with a wealth of material. The many sources cited are mostly from reasonably accessible journals (often from the MAA journals), and indicate a great deal of conscientious work on Barbeau's part. This is an excellent source to have on your shelves alongside Joe Roberts' Lure of the Integersand the John Conway and Richard Guy's >Book of Numbers for those times when you want to have some serious fun with numbers.

David Graves (dgraves@elmira.edu) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in opera and history of astronomy.