Fibonacci and Lucas Numbers with Applications / Edition 1

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Overview

Fibonacci and Lucas numbers are two of the most fascinating number sequences in mathematics. There is only a handful of books on the subject matter, these neither authoritative nor fully reference in nature.
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Editorial Reviews

From the Publisher
"…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 numbers...an 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)

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Product Details

Meet the Author

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.
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Table of Contents

Preface
List of Symbols
1 Leonardo Fibonacci 1
2 The Rabbit Problem 4
3 Fibonacci Numbers in Nature 16
4 Fibonacci Numbers: Additional Occurrences 51
5 Fibonacci and Lucas Identities 69
6 Geometric Paradoxes 100
7 Generalized Fibonacci Numbers 109
8 Additional Fibonacci and Lucas Formulas 116
9 The Euclidean Algorithm 132
10 Solving Recurrence Relations 142
11 Completeness Theorems 147
12 Pascal's Triangle 151
13 Pascal-Like Triangles 164
14 Additional Pascal-Like Triangles 180
15 Hosoya's Triangle 187
16 Divisibility Properties 196
17 Generalized Fibonacci Numbers Revisited 211
18 Generating Functions 215
19 Generating Functions Revisited 227
20 The Golden Ratio 239
21 The Golden Ratio Revisited 248
22 Golden Triangles 267
23 Golden Rectangles 273
24 Fibonacci Geometry 294
25 Regular Pentagons 308
26 The Golden Ellipse and Hyperbola 328
27 Continued Fractions 332
28 Weighted Fibonacci and Lucas Sums 340
29 Fibonacci and Lucas Sums Revisited 349
30 The Knapsack Problem 356
31 Fibonacci Magic Squares 360
32 Fibonacci Matrices 362
33 Fibonacci Determinants 387
34 Fibonacci and Lucas Congruences 402
35 Fibonacci and Lucas Periodicity 415
36 Fibonacci and Lucas Series 424
37 Fibonacci Polynomials 443
38 Lucas Polynomials 459
39 Jacobsthal Polynomials 469
40 Zeros of Fibonacci and Lucas Polynomials 477
41 Morgan-Voyce Polynomials 480
42 Fibonometry 496
43 Fibonacci and Lucas Subscripts 511
44 Gaussian Fibonacci and Lucas Numbers 518
45 Analytic Extensions 523
46 Tribonacci Numbers 527
47 Tribonacci Polynomials 533
App. A.1: Fundamentals 537
App. A.2 The First 100 Fibonacci and Lucas Numbers 553
App. A.3 The First 100 Fibonacci Numbers and Their Prime Factorizations 556
App. A.4 The First 100 Lucas Numbers and Their Prime Factorizations 559
References 562
Solutions to Odd-Numbered Exercises 577
Index 641
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