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Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

Book 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

Table of Contents

  1. COVER
  2. TITLE PAGE
  3. COPYRIGHT
  4. DEDICATION
  5. LIST OF SYMBOLS
  6. PREFACE
  7. CHAPTER 1: LEONARDO FIBONACCI
  8. CHAPTER 2: FIBONACCI NUMBERS
    1. 2.1 FIBONACCI'S RABBITS
    2. 2.2 FIBONACCI NUMBERS
    3. 2.3 FIBONACCI AND LUCAS CURIOSITIES
  9. CHAPTER 3: FIBONACCI NUMBERS IN NATURE
    1. 3.1 FIBONACCI, FLOWERS, AND TREES
    2. 3.2 FIBONACCI AND MALE BEES
    3. 3.3 FIBONACCI, LUCAS, AND SUBSETS
    4. 3.4 FIBONACCI AND SEWAGE TREATMENT
    5. 3.5 FIBONACCI AND ATOMS
    6. 3.6 FIBONACCI AND REFLECTIONS
    7. 3.7 PARAFFINS AND CYCLOPARAFFINS
    8. 3.8 FIBONACCI AND MUSIC
    9. 3.9 FIBONACCI AND POETRY
    10. 3.10 FIBONACCI AND NEUROPHYSIOLOGY
    11. 3.11 ELECTRICAL NETWORKS
  10. CHAPTER 4: ADDITIONAL FIBONACCI AND LUCAS OCCURRENCES
    1. 4.1 FIBONACCI OCCURRENCES
    2. 4.2 FIBONACCI AND COMPOSITIONS
    3. 4.3 FIBONACCI AND PERMUTATIONS
    4. 4.4 FIBONACCI AND GENERATING SETS
    5. 4.5 FIBONACCI AND GRAPH THEORY
    6. 4.6 FIBONACCI WALKS
    7. 4.7 FIBONACCI TREES
    8. 4.9 FIBONACCI AND THE STOCK MARKET
  11. CHAPTER 5: FIBONACCI AND LUCAS IDENTITIES
    1. 5.1 SPANNING TREE OF A CONNECTED GRAPH
    2. 5.2 BINET'S FORMULAS
    3. 5.3 CYCLIC PERMUTATIONS AND LUCAS NUMBERS
    4. 5.4 COMPOSITIONS REVISITED
    5. 5.5 NUMBER OF DIGITS IN AND
    6. 5.6 THEOREM 5.8 REVISITED
    7. 5.7 CATALAN'S IDENTITY
    8. 5.8 ADDITIONAL FIBONACCI AND LUCAS IDENTITIES
    9. 5.9 FERMAT AND FIBONACCI
    10. 5.10 FIBONACCI AND
  12. CHAPTER 6: GEOMETRIC ILLUSTRATIONS AND PARADOXES
    1. 6.1 GEOMETRIC ILLUSTRATIONS
    2. 6.2 CANDIDO'S IDENTITY
    3. 6.3 FIBONACCI TESSELLATIONS
    4. 6.4 LUCAS TESSELLATIONS
    5. 6.5 GEOMETRIC PARADOXES
    6. 6.6 CASSINI-BASED PARADOXES
    7. 6.7 ADDITIONAL PARADOXES
  13. CHAPTER 7: GIBONACCI NUMBERS
    1. 7.1 GIBONACCI NUMBERS
    2. 7.2 GERMAIN'S IDENTITY
  14. CHAPTER 8: ADDITIONAL FIBONACCI AND LUCAS FORMULAS
    1. 8.1 NEW EXPLICIT FORMULAS
    2. 8.2 ADDITIONAL FORMULAS
  15. CHAPTER 9: THE EUCLIDEAN ALGORITHM
    1. 9.1 THE EUCLIDEAN ALGORITHM
    2. 9.2 FORMULA (5.5) REVISITED
    3. 9.3 LAMé'S THEOREM
  16. CHAPTER 10: DIVISIBILITY PROPERTIES
    1. 10.1 FIBONACCI DIVISIBILITY
    2. 10.2 LUCAS DIVISIBILITY
    3. 10.3 FIBONACCI AND LUCAS RATIOS
    4. 10.4 AN ALTERED FIBONACCI SEQUENCE
  17. CHAPTER 11: PASCAL'S TRIANGLE
    1. 11.1 BINOMIAL COEFFICIENTS
    2. 11.2 PASCAL'S TRIANGLE
    3. 11.3 FIBONACCI NUMBERS AND PASCAL'S TRIANGLE
    4. 11.4 ANOTHER EXPLICIT FORMULA FOR
    5. 11.5 CATALAN'S FORMULA
    6. 11.6 ADDITIONAL IDENTITIES
    7. 11.7 FIBONACCI PATHS OF A ROOK ON A CHESSBOARD
  18. CHAPTER 12: PASCAL-LIKE TRIANGLES
    1. 12.1 SUMS OF LIKE-POWERS
    2. 12.2 AN ALTERNATE FORMULA FOR
    3. 12.3 DIFFERENCES OF LIKE-POWERS
    4. 12.4 CATALAN'S FORMULA REVISITED
    5. 12.5 A LUCAS TRIANGLE
    6. 12.6 POWERS OF LUCAS NUMBERS
    7. 12.7 VARIANTS OF PASCAL'S TRIANGLE
  19. CHAPTER 13: RECURRENCES AND GENERATING FUNCTIONS
    1. 13.1 LHRWCCs
    2. 13.2 GENERATING FUNCTIONS
    3. 13.3 A GENERATING FUNCTION FOR
    4. 13.4 A GENERATING FUNCTION FOR
    5. 13.5 SUMMATION FORMULA (5.1) REVISITED
    6. 13.6 A LIST OF GENERATING FUNCTIONS
    7. 13.7 COMPOSITIONS REVISITED
    8. 13.8 EXPONENTIAL GENERATING FUNCTIONS
    9. 13.9 HYBRID IDENTITIES
    10. 13.10 IDENTITIES USING THE DIFFERENTIAL OPERATOR
  20. CHAPTER 14: COMBINATORIAL MODELS I
    1. 14.1 A FIBONACCI TILING MODEL
    2. 14.2 A CIRCULAR TILING MODEL
    3. 14.3 PATH GRAPHS REVISITED
    4. 14.4 CYCLE GRAPHS REVISITED
    5. 14.5 TADPOLE GRAPHS
  21. CHAPTER 15: HOSOYA'S TRIANGLE
    1. 15.1 RECURSIVE DEFINITION
    2. 15.2 A MAGIC RHOMBUS
  22. CHAPTER 16: THE GOLDEN RATIO
    1. 16.1 RATIOS OF CONSECUTIVE FIBONACCI NUMBERS
    2. 16.2 THE GOLDEN RATIO
    3. 16.3 GOLDEN RATIO AS NESTED RADICALS
    4. 16.4 NEWTON'S APPROXIMATION METHOD
    5. 16.5 THE UBIQUITOUS GOLDEN RATIO
    6. 16.6 HUMAN BODY AND THE GOLDEN RATIO
    7. 16.7 VIOLIN AND THE GOLDEN RATIO
    8. 16.8 ANCIENT FLOOR MOSAICS AND THE GOLDEN RATIO
    9. 16.9 GOLDEN RATIO IN AN ELECTRICAL NETWORK
    10. 16.10 GOLDEN RATIO IN ELECTROSTATICS
    11. 16.11 GOLDEN RATIO BY ORIGAMI
    12. 16.12 DIFFERENTIAL EQUATIONS
    13. 16.13 GOLDEN RATIO IN ALGEBRA
    14. 16.14 GOLDEN RATIO IN GEOMETRY
  23. CHAPTER 17: GOLDEN TRIANGLES AND RECTANGLES
    1. 17.1 GOLDEN TRIANGLE
    2. 17.2 GOLDEN RECTANGLES
    3. 17.3 THE PARTHENON
    4. 17.4 HUMAN BODY AND THE GOLDEN RECTANGLE
    5. 17.5 GOLDEN RECTANGLE AND THE CLOCK
    6. 17.6 STRAIGHTEDGE AND COMPASS CONSTRUCTION
    7. 17.7 RECIPROCAL OF A RECTANGLE
    8. 17.8 LOGARITHMIC SPIRAL
    9. 17.9 GOLDEN RECTANGLE REVISITED
    10. 17.10 SUPERGOLDEN RECTANGLE
  24. CHAPTER 18: FIGEOMETRY
    1. 18.1 THE GOLDEN RATIO AND PLANE GEOMETRY
    2. 18.2 THE CROSS OF LORRAINE
    3. 18.3 FIBONACCI MEETS APOLLONIUS
    4. 18.4 A FIBONACCI SPIRAL
    5. 18.5 REGULAR PENTAGONS
    6. 18.6 TRIGONOMETRIC FORMULAS FOR
    7. 18.7 REGULAR DECAGON
    8. 18.8 FIFTH ROOTS OF UNITY
    9. 18.9 A PENTAGONAL ARCH
    10. 18.10 REGULAR ICOSAHEDRON AND DODECAHEDRON
    11. 18.11 GOLDEN ELLIPSE
    12. 18.12 GOLDEN HYPERBOLA
  25. CHAPTER 19: CONTINUED FRACTIONS
    1. 19.1 FINITE CONTINUED FRACTIONS
    2. 19.2 CONVERGENTS OF A CONTINUED FRACTION
    3. 19.3 INFINITE CONTINUED FRACTIONS
    4. 19.4 A NONLINEAR DIOPHANTINE EQUATION
  26. CHAPTER 20: FIBONACCI MATRICES
    1. 20.1 THE Q-MATRIX
    2. 20.2 EIGENVALUES OF
    3. 20.3 FIBONACCI AND LUCAS VECTORS
    4. 20.4 AN INTRIGUING FIBONACCI MATRIX
    5. 20.5 AN INFINITE-DIMENSIONAL LUCAS MATRIX
    6. 20.6 AN INFINITE-DIMENSIONAL GIBONACCI MATRIX
    7. 20.7 THE LAMBDA FUNCTION
  27. CHAPTER 21: GRAPH-THEORETIC MODELS I
    1. 21.1 A GRAPH-THEORETIC MODEL FOR FIBONACCI NUMBERS
    2. 21.2 BYPRODUCTS OF THE COMBINATORIAL MODELS
    3. 21.3 SUMMATION FORMULAS
  28. CHAPTER 22: FIBONACCI DETERMINANTS
    1. 22.1 AN APPLICATION TO GRAPH THEORY
    2. 22.2 THE SINGULARITY OF FIBONACCI MATRICES
    3. 22.3 FIBONACCI AND ANALYTIC GEOMETRY
  29. CHAPTER 23: FIBONACCI AND LUCAS CONGRUENCES
    1. 23.1 FIBONACCI NUMBERS ENDING IN ZERO
    2. 23.2 LUCAS NUMBERS ENDING IN ZERO
    3. 23.3 ADDITIONAL CONGRUENCES
    4. 23.4 LUCAS SQUARES
    5. 23.5 FIBONACCI SQUARES
    6. 23.6 A GENERALIZED FIBONACCI CONGRUENCE
    7. 23.7 FIBONACCI AND LUCAS PERIODICITIES
    8. 23.8 LUCAS SQUARES REVISITED
    9. 23.9 PERIODICITIES MODULO
  30. CHAPTER 24: FIBONACCI AND LUCAS SERIES
    1. 24.1 A FIBONACCI SERIES
    2. 24.2 A LUCAS SERIES
    3. 24.3 FIBONACCI AND LUCAS SERIES REVISITED
    4. 24.4 A FIBONACCI POWER SERIES
    5. 24.5 GIBONACCI SERIES
    6. 24.6 ADDITIONAL FIBONACCI SERIES
  31. CHAPTER 25: WEIGHTED FIBONACCI AND LUCAS SUMS
    1. 25.1 WEIGHTED SUMS
    2. 25.2 GAUTHIER'S DIFFERENTIAL METHOD
  32. CHAPTER 26: FIBONOMETRY I
    1. 26.1 GOLDEN RATIO AND INVERSE TRIGONOMETRIC FUNCTIONS
    2. 26.2 GOLDEN TRIANGLE REVISITED
    3. 26.3 GOLDEN WEAVES
    4. 26.4 ADDITIONAL FIBONOMETRIC BRIDGES
    5. 26.5 FIBONACCI AND LUCAS FACTORIZATIONS
  33. CHAPTER 27: COMPLETENESS THEOREMS
    1. 27.1 COMPLETENESS THEOREM
    2. 27.2 EGYPTIAN ALGORITHM FOR MULTIPLICATION
  34. CHAPTER 28: THE KNAPSACK PROBLEM
    1. 28.1 THE KNAPSACK PROBLEM
  35. CHAPTER 29: FIBONACCI AND LUCAS SUBSCRIPTS
    1. 29.1 FIBONACCI AND LUCAS SUBSCRIPTS
    2. 29.2 GIBONACCI SUBSCRIPTS
    3. 29.3 A RECURSIVE DEFINITION OF
  36. CHAPTER 30: FIBONACCI AND THE COMPLEX PLANE
    1. 30.1 GAUSSIAN NUMBERS
    2. 30.2 GAUSSIAN FIBONACCI AND LUCAS NUMBERS
    3. 30.3 ANALYTIC EXTENSIONS
  37. APPENDIX 1: FUNDAMENTALS
    1. SEQUENCES
    2. SUMMATION AND PRODUCT NOTATIONS
    3. INDEXED SUMMATION
    4. THE PRODUCT NOTATION
    5. THE FACTORIAL NOTATION
    6. FLOOR AND CEILING FUNCTIONS
    7. THE WELL-ORDERING PRINCIPLE (WOP)
    8. MATHEMATICAL INDUCTION
    9. SUMMATION FORMULAS
    10. RECURSION
    11. RECURSIVE DEFINITION OF A FUNCTION
    12. THE DIVISION ALGORITHM
    13. DIV AND MOD OPERATORS
    14. DIVISIBILITY RELATION
    15. DIVISIBILITY PROPERTIES
    16. PIGEONHOLE PRINCIPLE
    17. ADDITION PRINCIPLE
    18. UNION AND INTERSECTION
    19. GCD AND LCM
    20. GREATEST COMMON DIVISOR
    21. A SYMBOLIC DEFINITION OF GCD
    22. RELATIVELY PRIME INTEGERS
    23. GCD OF POSITIVE INTEGERS
    24. FUNDAMENTAL THEOREM OF ARITHMETIC
    25. CANONICAL DECOMPOSITION
    26. LEAST COMMON MULTIPLE
    27. A SYMBOLIC DEFINITION OF LCM
    28. MATRICES AND DETERMINANTS
    29. MATRICES
    30. EQUALITY OF MATRICES
    31. ZERO AND IDENTITY MATRICES
    32. MATRIX OPERATIONS
    33. MATRIX MULTIPLICATION
    34. DETERMINANTS
    35. MINORS AND COFACTORS
    36. DETERMINANT OF A SQUARE MATRIX
    37. CONGRUENCES
    38. CONGRUENCE MODULO
  38. APPENDIX 2: THE FIRST 100 FIBONACCI AND LUCAS NUMBERS
  39. APPENDIX 3: THE FIRST 100 FIBONACCI NUMBERS AND THEIR PRIME FACTORIZATIONS
  40. APPENDIX 4: THE FIRST 100 LUCAS NUMBERS AND THEIR PRIME FACTORIZATIONS
  41. ABBREVIATIONS
  42. REFERENCES
  43. SOLUTIONS TO ODD-NUMBERED EXERCISES
  44. INDEX
  45. PURE AND APPLIED MATHEMATICS
  46. End User License Agreement