The first comprehensive survey of mathematics' most fascinating number sequencesFibonacci 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 matricesFibonacci 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.
Author(s): Thomas Koshy
Edition: 1
Publisher: Wiley-Interscience
Year: 2001
Language: English
Pages: 676
Tags: Математика;Теория чисел;
Fibonacci and Lucas Numbers with Applications......Page 5
CONTENTS......Page 9
Preface......Page 13
List of Symbols......Page 17
1. Leonardo Fibonacci......Page 21
2. The Rabbit Problem......Page 24
3. Fibonacci Numbers in Nature......Page 36
4. Fibonacci Numbers: Additional Occurrences......Page 71
5. Fibonacci and Lucas Identities......Page 89
6. Geometric Paradoxes......Page 120
7. Generalized Fibonacci Numbers......Page 129
8. Additional Fibonacci and Lucas Formulas......Page 136
9. The Euclidean Algorithm......Page 152
10. Solving Recurrence Relations......Page 162
11. Completeness Theorems......Page 167
12. Pascal's Triangle......Page 171
13. Pascal-Like Triangles......Page 184
14. Additional Pascal-Like Triangles......Page 200
15. Hosoya's Triangle......Page 207
16. Divisibility Properties......Page 216
17. Generalized Fibonacci Numbers Revisited......Page 231
18. Generating Functions......Page 235
19. Generating Functions Revisited......Page 247
20. The Golden Ratio......Page 259
21. The Golden Ratio Revisited......Page 268
22. Golden Triangles......Page 287
23. Golden Rectangles......Page 293
24. Fibonacci Geometry......Page 314
25. Regular Pentagons......Page 328
26. The Golden Ellipse and Hyperbola......Page 348
27. Continued Fractions......Page 352
28. Weighted Fibonacci and Lucas Sums......Page 360
29. Fibonacci and Lucas Sums Revisited......Page 369
30. The Knapsack Problem......Page 376
31. Fibonacci Magic Squares......Page 380
32. Fibonacci Matrices......Page 382
33. Fibonacci Determinants......Page 407
34. Fibonacci and Lucas Congruences......Page 422
35. Fibonacci and Lucas Periodicity......Page 435
36. Fibonacci and Lucas Series......Page 444
37. Fibonacci Polynomials......Page 463
38. Lucas Polynomials......Page 479
39. Jacobsthal Polynomials......Page 489
40. Zeros of Fibonacci and Lucas Polynomials......Page 497
41. Morgan-Voyce Polynomials......Page 500
42. Fibonometry......Page 516
43. Fibonacci and Lucas Subscripts......Page 531
44. Gaussian Fibonacci and Lucas Numbers......Page 538
45. Analytic Extensions......Page 543
46. Tribonacci Numbers......Page 547
47. Tribonacci Polynomials......Page 553
A.1. Fundamentals......Page 557
A.2. The First 100 Fibonacci and Lucas Numbers......Page 573
A.3. The First 100 Fibonacci Numbers and Their Prime Factorizations......Page 576
A.4. The First 100 Lucas Numbers and Their Prime Factorizations......Page 579
References......Page 582
Solutions to Odd-Numbered Exercises......Page 597
Index......Page 661
