Perfect Numbers and Fibonacci Sequences
English | 2022 | ISBN: 9811244073 | 261 pages | True PDF | 129.15 MB
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.
Contents:
- Preface
- The History of Perfect Numbers:[list]
- 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
- Properties of Even Perfect Numbers
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
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
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 π L (n)
Prime Divisors of Lucas Numbers Perfect Numbers and Fibonacci Primes:
- Square Sum Perfect Numbers
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
Prime Factorization of the First 100 Lucas Numbers Bibliography
Index
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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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