Fibonacci and Lucas Numbers with Applications
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The first comprehensive survey of mathematics' most fascinating number sequences
Fibonacci and Lucas numbers have intrigued amateur and professional mathematicians for centuries. This volume represents the first attempt to compile a definitive history and authoritative analysis of these famous integer sequences, complete with a wealth of exciting applications, enlightening examples, and fun exercises that offer numerous opportunities for exploration and experimentation.
The author has assembled a myriad of fascinating properties of both Fibonacci and Lucas numbers-as developed by a wide range of sources-and catalogued their applications in a multitude of widely varied disciplines such as art, stock market investing, engineering, and neurophysiology. Most of the engaging and delightful material here is easily accessible to college and even high school students, though advanced material is included to challenge more sophisticated Fibonacci enthusiasts. A historical survey of the development of Fibonacci and Lucas numbers, biographical sketches of intriguing personalities involved in developing the subject, and illustrative examples round out this thorough and amusing survey. Most chapters conclude with numeric and theoretical exercises that do not rely on long and tedious proofs of theorems. Highlights include:
* Balanced blend of theory and real-world applications
* Excellent reference material for student reports and projects
* User-friendly, informal, and entertaining writing style
* Historical interjections and short biographies that add a richer perspective to the topic
* Reference sections providing important symbols, problem solutions, and fundamental properties from the theory of numbers and matrices
Fibonacci and Lucas Numbers with Applications provides mathematicians with a wealth of reference material in one convenient volume and presents an in-depth and entertaining resource for enthusiasts at every level and from any background.


THOMAS KOSHY, Ph.D., is Professor of Mathematics at Framingham State College in Framingham, Massachusetts. He is the author of the forthcoming book Elementary Number Theory with Applications, among several other books, as well as numerous articles across a wide range of topics.



List of Symbols.

Leonardo Fibonacci.

The Rabbit Problem.

Fibonacci Numbers in Nature.

Fibonacci Numbers: Additional Occurrances.

Fibonacci and Lucas Identities.

Geometric Paradoxes.

Generalized Fibonacci Numbers.

Additional Fibonacci and Lucas Formulas.

The Euclidean Algorithm.

Solving Recurrence Relations.

Completeness Theorems.

Pascal's Triangle.

Pascal-Like Triangles.

Additional Pascal-Like Triangles.

Hosoya's Triangle.

Divisibility Properties.

Generalized Fibonacci Numbers Revisited.

Generating Functions.

Generating Functions Revisited.

The Golden Ratio.

The Golden Ratio Revisited.

Golden Triangles.

Golden Rectangles.

Fibonacci Geometry.

Regular Pentagons.

The Golden Ellipse and Hyperbola.

Continued Fractions.

Weighted Fibonacci and Lucas Sums.

Weighted Fibonacci and Lucas Sums Revisited.

The Knapsack Problem.

Fibonacci Magic Squares.

Fibonacci Matrices.

Fibonacci Determinants.

Fibonacci and Lucas Congruences.

Fibonacci and Lucas Periodicity.

Fibonacci and Lucas Series.

Fibonacci Polynomials.

Lucas Polynomials.

Jacobsthal Polynomials.

Zeros of Fibonacci and Lucas Polynomials.

Morgan-Voyce Polynomials.


Fibonacci and Lucas Subscripts.

Gaussian Fibonacci and Lucas Numbers.

Analytic Extensions.

Tribonacci Numbers.

Tribonacci Polynomials. Appendix 1: Fundamentals.

Appendix 2: The First 100 Fibonacci and Lucas Numbers.

Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations.

Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations.


Solutions to Odd-Numbered Exercises.



"...a 'must' for fans of Fibonacci..." (Translated from French, Bulletin AMQ, October 2005)

"...presents a look at the numerous applications of Fibonacci and Lucas numbers in a range of disciplines." (SciTech Book News, Vol. 25, No. 4, December 2001)

"...presents the history, mathematics, and applications of the Fibonacci and Lucas sequences...An important feature of the book is the many...exercises..." (Mathematical Reviews, 2002f)

"...beautiful and well worth the reading...with many exercises and a good bibliography, this book will fascinate both students and teachers." (Mathematics Teacher, Vol. 95, No. 5, May 2002)

"A delightful romp through all things Fibonacci." (American Mathematical Monthly, January 2003)

"...a delightful book which should prove of great value...the most comprehensive collection of results, theorems, and references regarding Fibonacci numbers and their applications to date..." (The Fibonacci Quarterly, February 2002)

"...a splendid compendium of everything that most of us will ever need to know about the Fibonacci and Lucas invaluable reference for experts and non-experts alike..." (The Mathematical Gazette)

"...a definitive history and authoritative analysis ...a myriad of fascinating properties of both Fibonacci and Lucas numbers..." (Mathematical Didactics, 2003) [Koshy's] book is without doubt the most comprehensive and scholarly work on Fibonacci numbers to date and I am sure that it will quickly signal its presence and impose itself as an authoritative reference manual on Fibonacci numbers.
--Napoleon Gauthier, Royal Military College of Canada, Kingston, ON

What a gem this is! [...] My only regrest about the book is that it wasn't around years ago. It fills such a void.
--Monte Zerger, Adams State College, Alamosa, CO