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Fibonacci and Lucas Numbers with Applications
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The first 100 Lucas Numbers and their prime factorizations were given in this article, where they were shown to be a special case of the first 100 Fibonacci Numbers and Lucas Polynomials.Abstract:
Preface. 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. Fibonometry. 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. References. Solutions to Odd-Numbered Exercises. Index.read more
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On a conjecture of broughan
TL;DR: In this paper, the authors confirm a conjecture of Broughan, concerning the closure of the set of Fibonacci numbers in the full topology over Z, and show that the conjecture is correct.
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Fibonacci-zeta Infinite Series Associated with the Polygamma Functions
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Hyperbolic Tribonacci and Tribonacci-Lucas sequences
TL;DR: In this article, the authors define hyperbolic Tribonacci and tethered-Lucas numbers and obtain recurrence relations, Binet's formulas and generating functions for these sequences, and find sums formulas involving odd and even terms of these sequences.
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The spectral norm of a Horadam circulant matrix
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A note on annular bound for the zeros of a polynomial
TL;DR: In this paper, the annular bound for the zeros of a polynomial based on the identities related to the generalized Fibonacci sequence with arbitrary initial condition was shown.
References
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Elementary number theory
TL;DR: The book as mentioned in this paper is designed for a first course in number theory with minimal prerequisites and is designed to stimulate curiosity about numbers and their properties, including almost a thousand imaginative exercises and problems.
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Unsolved problems in number theory
TL;DR: The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.
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Linear Algebra With Applications
TL;DR: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra.