# Fibonacci and Lucas Numbers with Applications, Volume 1

Wiley Series in Pure and Applied Mathematics (Series Nr. 1)

2. Edition January 2018

704 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-118-74212-9**

### Short Description

Praise for the First Edition

" ...beautiful and well worth the reading ... with many exercises and a good bibliography, this book will fascinate both students and teachers." Mathematics Teacher

Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment.

In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The Second Edition offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features:

* A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio

* Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication

* Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers

* A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology

The Second Edition is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers.

Thomas Koshy, PhD, is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University.

"Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications." Marjorie Bicknell-Johnson

Praise for the First Edition

" ...beautiful and well worth the reading ... with many exercises and a good bibliography, this book will fascinate both students and teachers." Mathematics Teacher

Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment.

In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The Second Edition offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features:

* A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio

* Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication

* Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers

* A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology

The Second Edition is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers.

Thomas Koshy, PhD, is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University.

"Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications." Marjorie Bicknell-Johnson

2 Fibonacci Numbers 13

2.1 Fibonacci's Rabbits 13

2.2 Fibonacci Numbers 14

2.3 Fibonacci and Lucas Curiosities 17

3 Fibonacci Numbers in Nature 27

3.1 Fibonacci, Flowers, and Trees 28

3.2 Fibonacci and Male Bees 31

3.3 Fibonacci, Lucas, and Subsets 32

3.4 Fibonacci and Sewage Treatment 34

3.5 Fibonacci and Atoms 35

3.6 Fibonacci and Reflections 36

3.7 Paraffins and Cycloparaffins 38

3.8 Fibonacci and Music 41

3.9 Fibonacci and Poetry 42

3.10 Fibonacci and Neurophysiology 43

3.11 Electrical Networks 45

4 Additional Fibonacci and Lucas Occurrences 53

4.1 Fibonacci Occurrences 53

4.2 Fibonacci and Compositions 58

4.3 Fibonacci and Permutations 61

4.4 Fibonacci and Generating Sets 63

4.5 Fibonacci and Graph Theory 64

4.6 Fibonacci Walks 66

4.7 Fibonacci Trees 68

4.8 Partitions 71

4.9 Fibonacci and the Stock Market 72

5 Fibonacci and Lucas Identities 77

5.1 Spanning Tree of a Connected Graph 79

5.2 Binet's Formulas 83

5.3 Cyclic Permutations and Lucas Numbers 91

5.4 Compositions Revisited 94

5.5 Number of Digits in Fn and Ln 94

5.6 Theorem 5.8 Revisited 95

5.7 Catalan's Identity 99

5.8 Additional Fibonacci and Lucas Identities 102

5.9 Fermat and Fibonacci 108

5.10 Fibonacci and 110

6 Geometric Illustrations and Paradoxes 117

6.1 Geometric Illustrations 117

6.2 Candido's Identity 121

6.3 Fibonacci Tessellations 123

6.4 Lucas Tessellations 123

6.5 Geometric Paradoxes 124

6.6 Cassini-Based Paradoxes 124

6.7 Additional Paradoxes 129

7 Gibonacci Numbers 133

7.1 Gibonacci Numbers 133

7.2 Germain's Identity 139

8 Additional Fibonacci and Lucas Formulas 145

8.1 New Explicit Formulas 145

8.2 Additional Formulas 148

9 The Euclidean Algorithm 159

9.1 The Euclidean Algorithm 160

9.2 Formula (5.5) Revisited 162

9.3 Lamé's Theorem 164

10 Divisibility Properties 167

10.1 Fibonacci Divisibility 167

10.2 Lucas Divisibility 173

10.3 Fibonacci and Lucas Ratios 173

10.4 An Altered Fibonacci Sequence 178

11 Pascal's Triangle 185

11.1 Binomial Coefficients 185

11.2 Pascal's Triangle 186

11.3 Fibonacci Numbers and Pascal's Triangle 188

11.4 Another Explicit Formula for Ln 191

11.5 Catalan's Formula 192

11.6 Additional Identities 192

11.7 Fibonacci Paths of a Rook on a Chessboard 194

12 Pascal-like Triangles 199

12.1 Sums of Like-Powers 199

12.2 An Alternate Formula for Ln 202

12.3 Differences of Like-Powers 202

12.4 Catalan's Formula Revisited 204

12.5 A Lucas Triangle 205

12.6 Powers of Lucas Numbers 209

12.7 Variants of Pascal's Triangle 211

13 Recurrences and Generating Functions 219

13.1 LHRWCCs 219

13.2 Generating Functions 223

13.3 A Generating Function For F3n 233

13.4 A Generating Function For F3 n 234

13.5 Summation Formula (5.1) Revisited 234

13.6 A List of Generating Functions 235

13.7 Compositions Revisited 238

13.8 Exponential Generating Functions 239

13.9 Hybrid Identities 241

13.10Identities Using the Differential Operator 242

14 Combinatorial Models I 249

14.1 A Fibonacci Tiling Model 249

14.2 A Circular Tiling Model 255

14.3 Path Graphs Revisited 259

14.4 Cycle Graphs Revisited 262

14.5 Tadpole Graphs 263

15 Hosoya's Triangle 271

15.1 Recursive Definition 271

15.2 A Magic Rhombus 273

16 The Golden Ratio 279

16.1 Ratios of Consecutive Fibonacci Numbers 279

16.2 The Golden Ratio 281

16.3 Golden Ratio as Nested Radicals 285

16.4 Newton's Approximation Method 286

16.5 The Ubiquitous Golden Ratio 288

16.6 Human Body and the Golden Ratio 289

16.7 Violin and the Golden Ratio 290

16.8 Ancient Floor Mosaics and the Golden Ratio 290

16.9 Golden Ratio in an Electrical Network 290

16.10Golden Ratio in Electrostatics 291

16.11Golden Ratio by Origami 292

16.12Differential Equations 297

16.13Golden Ratio in Algebra 299

16.14Golden Ratio in Geometry 300

17 Golden Triangles and Rectangles 309

17.1 Golden Triangle 309

17.2 Golden Rectangles 314

17.3 The Parthenon 317

17.4 Human Body and the Golden Rectangle 318

17.5 Golden Rectangle and the Clock 319

17.6 Straightedge and Compass Construction 320

17.7 Reciprocal of a Rectangle 321

17.8 Logarithmic Spiral 322

17.9 Golden Rectangle Revisited 324

17.10Supergolden Rectangle 324

18 Figeometry 329

18.1 The Golden Ratio and Plane Geometry 329

18.2 The Cross of Lorraine 335

18.3 Fibonacci Meets Appollonius 337

18.4 A Fibonacci Spiral 338

18.5 Regular Pentagons 339

18.6 Trigonometric Formulas for Fn 343

18.7 Regular Decagon 347

18.8 Fifth Roots of Unity 348

18.9 A Pentagonal Arch 351

18.10 Regular Icosahedron and Dodecahedron 351

18.11 Golden Ellipse 352

18.12 Golden Hyperbola 354

19 Continued Fractions 361

19.1 Finite Continued Fractions 361

19.2 Convergents of a Continued Fraction 364

19.3 Infinite Continued Fractions 366

19.4 A Nonlinear Diophantine Equation 368

20 Fibonacci Matrices 371

20.1 The Q-Matrix 371

20.2 Eigenvalues of Qn 378

20.3 Fibonacci and Lucas Vectors 384

20.4 An Intriguing Fibonacci Matrix 386

20.5 An Infinite-Dimensional Lucas Matrix 391

20.6 An Infinite-Dimensional Gibonacci Matrix 397

20.7 The Lambda Function 398

21 Graph-theoretic Models I 407

21.1 A Graph-theoretic Model for Fibonacci Numbers 407

21.2 Byproducts of the Combinatorial Models 409

21.3 Summation Formulas 415

22 Fibonacci Determinants 419

22.1 An Application to Graph Theory 419

22.2 The Singularity of Fibonacci Matrices 425

22.3 Fibonacci and Analytic Geometry 427

23 Fibonacci and Lucas Congruences 437

23.1 Fibonacci Numbers Ending in Zero 437

23.2 Lucas Numbers Ending in Zero 437

23.3 Additional Congruences 438

23.4 Lucas Squares 439

23.5 Fibonacci Squares 440

23.6 A Generalized Fibonacci Congruence 442

23.7 Fibonacci and Lucas Periodicities 449

23.8 Lucas Squares Revisited 450

23.9 Periodicities Modulo 10n 452

24 Fibonacci and Lucas Series 461

24.1 A Fibonacci Series 461

24.2 A Lucas Series 463

24.3 Fibonacci and Lucas Series Revisited 464

24.4 A Fibonacci Power Series 467

24.5 Gibonacci Series 472

24.6 Additional Fibonacci Series 474

25 Weighted Fibonacci and Lucas Sums 481

25.1 Weighted Sums 481

25.2 Gauthier's Differential Method 488

26 Fibonometry I 495

26.1 Golden Ratio and Inverse Trigonometric Functions 495

26.2 Golden Triangle Revisited 496

26.3 Golden Weaves 497

26.4 Additional Fibonometric Bridges 498

26.5 Fibonacci and Lucas Factorizations 504

27 Completeness Theorems 509

27.1 Completeness Theorem 509

27.2 Egyptian Algorithm for Multiplication 510

28 The Knapsack Problem 513

28.1 The Knapsack Problem 513

29 Fibonacci and Lucas Subscripts 517

29.1 Fibonacci and Lucas Subscripts 517

29.2 Gibonacci Subscripts 519

29.3 A Recursive Definition of Yn 520

30 Fibonacci and the Complex Plane 525

30.1 Gaussian Numbers 525

30.2 Gaussian Fibonacci and Lucas Numbers 526

30.3 Analytic Extensions 530

1 A.1 Fundamentals 537

SOLUTIONS TO ODD-NUMBERED EXERCISES 575