PERFECT NUMBERS AND FIBONACCI SEQUENCES
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In this book, we first review the history and current situation of the perfect number problem, including the origin story of the Mersenne primes, and then consider the history and current situation of the Fibonacci sequence. Both topics include results from our own research. In the later sections, we define the square sum perfect numbers, and describe for the first time the secret relationships connecting the square sum perfect numbers, the Fibonacci sequence, the Lucas sequence, the twin prime conjecture, and the Fermat primes. Throughout, we raise various interesting questions and conjectures.
Sample Chapter(s)
Preface
Chapter 1: The History of Perfect Numbers
Contents:
Preface
The History of Perfect Numbers:
What are Perfect Numbers?
Euclid's Elements
Nicomachus
Sums of Squares and Cubes
Ibn al-Haytham
Mersenne Numbers and Mersenne Primes
Fermat and Descartes
The Euclid–Euler Theorem
Ivan Pervushin
The Lucas–Lehmer Primality Test
The Great Internet Mersenne Prime Search
Questions Related to Perfect Numbers:
Properties of Even Perfect Numbers
Open Questions
Odd Perfect Numbers
Touchard's Theorem
Deficient and Abundant Numbers
Weird Numbers and Semiperfect Numbers
Ore Numbers and the Harmonic Mean
Variations on Ore Numbers
Amicable Numbers
Multiply Perfect Numbers
Three Further Generalizations
S-perfect Numbers
The Golden Ratio Conjecture
The Fibonacci Sequence:
Fibonacci, Leonardo of Pisa
The Rabbit Problem
General Terms and Limits
Connection with Continued Fractions
Three Identities
Equations Between Binomial Coefficients
Divisibility Sequences
Zeckendorf's Theorem
From Base 2 to Base 3
Hilbert's Tenth Problem
Lucas Numbers and Lucas Sequences:
The Lucas Numbers
Criteria for Fibonacci Numbers
Prime Divisors of Fibonacci Numbers
Fibonacci Congruences
A More General Congruence
Narayana Sequence Congruences
Pythagorean Triples
Diophantine m-tupples
Generating Functions
Lucas Sequences
Pisano Period
π(n) and πL(n)
Prime Divisors of Lucas Numbers
Perfect Numbers and Fibonacci Primes:
Square Sum Perfect Numbers
Some Lemmas
Proof of Theorems
A New Conjecture Concerning Perfect Numbers
Affine Square Sum Perfect Numbers
Square Sum Perfect Numbers and the Twin Prime Conjecture
Fermat Primes and GM Numbers
The abcd Equation
Applications of Elliptic Curves
Results with Lucas Sequences
Appendices:
The First 100 Fibonacci Numbers and Their Prime Factorizations
The First 100 Lucas Numbers and Their Prime Factorizations
Prime Factorization of the First 100 Lucas Numbers
Bibliography
Index
Readership: It is the first monograph to connecting perfect number and Fibonacci sequence, the two old mathematical problems, with rich history, it will attract number theorists and combinatorists in the world. Also good for maths teachers and fans.
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W21-Writing for 21st Century kids (Future & Perfect Tenses) (未來及完成式) (Book+APP)
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