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Fibonacci Numbers

Nicolai N. Vorobiev
Publication Date: 
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[Reviewed by
Raymond N. Greenwell
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This is a wonderful book with a huge amount of information on Fibonacci numbers. First published in the 1950s for high school students enrolled in a mathematical circle at the Leningrad State University, the book only assumes knowledge of high school mathematics, but the reader will need the sophistication of a good upper-level mathematics major. The warmth and enthusiasm of the late author comes through Mircea Martin's English translation. Given the popular awareness that the best selling novel The Da Vinci Code has given to the Fibonacci numbers and the golden ratio, the book under review comes at an opportune time for the mathematician interested in learning a few new nuggets of information about this topic.

The first chapter, "The Simplest Properties of Fibonacci Numbers," discusses the usual properties and identities, as well as some not-so-simple properties. For example, in the discussion of the sums of the reciprocals of the Fibonacci numbers, Vorobiev states that a closed expression for the sum is unknown, but that the sum was recently shown to be irrational. He gives a combinatorial interpretation of some Fibonacci identities using the "Bunny Problem," involving the number of ways a bunny can jump along a lane divided into cells. He does not, however, resort to combinatorial arguments very often, relying more often on induction and on results derived from previous results. Readers interested in more combinatorial proofs of Fibonacci identities should investigate Proofs That Really Count by Benjamin and Quinn.

The second chapter, on "Number-Theoretic Properties of Fibonacci Numbers," discusses divisibility properties of Fibonacci numbers, such as the fact that all the odd divisors of Fibonacci numbers with odd subscripts are of the form 4t + 1. The final 16 pages of the chapter are devoted to a proof that each Fibonacci number, with the exception of 1, 8, and 144, has a proper divisor, where a proper divisor is defined as a prime number that divides into the Fibonacci number but not into any smaller Fibonacci number. The reader should note that this is not the usual definition of proper divisor, but one gets used to such nonstandard terminology in a book that uses un rather than fn or Fn for the n-th Fibonacci number, and α rather than Φ for the golden ratio.

The third chapter, on "Fibonacci Numbers and Continued Fractions," contains a nice discussion of continued fractions in general. The focus of the chapter is on the continued fraction expression for the golden ratio, but Vorobiev takes an occasional detour. For example, he uses continued fractions to prove a result due to Hermite, that every divisor of an integer of the form a2 + 1 can be expressed as the sum of two perfect squares. The key result is Hurwitz's Theorem, which shows, roughly speaking, that the golden ratio is not approximated well by rational numbers. This and other results explain clearly and in detail what is meant by the common saying that the golden ratio is the most irrational of numbers. That saying is not repeated in this book; for one thing, the author shows that there are infinitely many numbers that are equally irrational, but that they are all equivalent, in a sense the author makes precise, to the golden ratio.

The fourth chapter, "Fibonacci Numbers and Geometry," discuss the familiar topics of the golden rectangle and triangle and the pentagram. The golden ratio is formally introduced in this chapter, even though it appeared anonymously in Chapter 1 and referred to by name in Chapter 3. The last part of the chapter is devoted to an analysis of a version of the game of Nim whose solution involves the Fibonacci numbers. The famous 64 = 65 puzzle is briefly discussed as well.

The fifth chapter, "Fibonacci Numbers and Search Theory," is devoted to the problem of finding the minimum value of a function that decreases and then increases on an interval. The author explains why Fibonacci numbers are involved in the most efficient search method possible. I found this chapter less interesting than the others in that so much space was devoted to just one idea.

This book would be useful to any mathematician with a desire to know more about the Fibonacci numbers. A student could use it for a reading project on the Fibonacci numbers, but the lack of exercises prevent it from being a useful textbook.

There are a few typographical errors, such as O1 in place of Q1 on p. 94, and "matther" in place of "matter" on p. 140. One small but frequent annoyance is an extra space before a period or comma, with the worst offense being a line on p. 130 that begins with a period belonging to the sentence ending on the previous line. An error involving the Fibonacci numeration system on p. 43 has 1, slightly shifted to the left, in place of 100. A similar error on the next page makes one wonder if this is not an error but bad notation. An occasional awkward translation interrupts an otherwise smooth reading (e.g. "Suppose we take a unit segment AB (see Figure 2) and want to break it into two pieces in such a way that the greater part is the mean proportional between the smaller part and the whole segment.") Another slight annoyance is that the book is printed on thick, heavy paper, making it weigh more than it should.

Overall, I enjoyed the book and highly recommend it.



  • Arthur T. Benjamin and Jennifer J. Quinn, Proofs That Really Count, Mathematical Association of America, 2003, ISBN 0-88385-333-7.

  • Dan Brown, The Da Vinci Code, Doubleday, 2003, ISBN 0-385-50420-9.

Raymond N. Greenwell ( is Professor of Mathematics at Hofstra University in Hempstead, New York. His research interests include applied mathematics and statistics, and he is coauthor of the texts Finite Mathematics and Calculus with Applications, both published by Addison Wesley.

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