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Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications

Steven Vajda
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a thorough and easy-to-follow study of the Fibonacci numbers and their close relatives the Lucas numbers and the generalized Fibonacci numbers (all satisfy the same recursion but have different initial values). The book has some coverage of number-theoretic properties and continued fractions of these numbers, but the bulk of the material deals with identities. The present volume is an unaltered reprint of the 1989 Ellis Horwood edition.

The book gets off to a slow start, and at times seems to be wandering around, until about one-third of the way through when the pace and mathematical sophistication pick up. The book generally assumes nothing beyond college algebra, but includes a lengthy appendix that develops any more-advanced topics that are needed; most of these are from number theory. The book also includes a very handy appendix that repeats all the numbered formulas that have occurred; nearly all of these are Fibonacci identities.

The primary proof methods are telescoping sums and induction, with Binet’s explicit formulas used to some extent. Generating functions are only used for one or two examples. The “application” portions of the book are skimpy and mostly deal with applications to other parts of mathematics. There’s no mention of biology or phyllotaxis, but happily also no mention of “patently cranky claims” (p. 139) about the golden ratio. There are a modest number of typographical errors, none of them confusing. The book uses τ rather than the more common φ for the golden ratio.

The present book has similar coverage and prerequisites to Vorobiev’s Fibonacci Numbers, although the latter book goes much deeper on some topics. Benjamin & Quinn’s Proofs That Really Count gives a very different approach to Fibonacci identities through combinatorial arguments; their book uses Vajda’s appendix of Fibonacci identities as a checklist for the thoroughness of coverage of their approach.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  1. Preface
  2. Introduction
  3. Background
  4. Relationships
  5. Fibonacci numbers and the Golden Section
  6. Fibonacci Series
  7. Divisibility properties
  8. Congruences and uniformity
  9. Continued fractions and convergents
  10. Fibonacci representation
  11. Search and games
  12. Hyperbolic functions and Fibonacci numbers
  13. Meta-Fibonacci sequences (a letter from B. W. Conolly)
  14. The Golden Section in the plane
  15. The Golden Section in three-dimensional space
  16. Appendix
  17. List of formulae
  18. References
  19. Table of Fn and Ln and their prime factors
  20. Index