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.
Author(s): Tianxin Cai
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 260
City: Singapore
Contents
Preface
1. The History of Perfect Numbers
1.1. What are Perfect Numbers?
1.2. Euclid’s Elements
1.3. Nicomachus
1.4. Sums of Squares and Cubes
1.5. Ibn al-Haytham
1.6. Mersenne Numbers and Mersenne Primes
1.7. Fermat and Descartes
1.8. The Euclid–Euler Theorem
1.9. Ivan Pervushin
1.10. The Lucas–Lehmer Primality Test
1.11. The Great Internet Mersenne Prime Search
2. Questions Related to Perfect Numbers
2.1. Properties of Even Perfect Numbers
2.2. Open Questions
2.3. Odd Perfect Numbers
2.4. Touchard’s Theorem
2.5. Deficient and Abundant Numbers
2.6. Weird Numbers and Semiperfect Numbers
2.7. Ore Numbers and the Harmonic Mean
2.8. Variations on Ore Numbers
2.9. Amicable Numbers
2.10. Multiply Perfect Numbers
2.11. Three Further Generalizations
2.12. S-perfect Numbers
2.13. The Golden Ratio Conjecture
3. The Fibonacci Sequence
3.1. Fibonacci, Leonardo of Pisa
3.2. The Rabbit Problem
3.3. General Terms and Limits
3.4. Connection with Continued Fractions
3.5. Three Identities
3.6. Equations Between Binomial Coefficients
3.7. Divisibility Sequences
3.8. Zeckendorf’s Theorem
3.9. From Base 2 to Base 3
3.10. Hilbert’s Tenth Problem
4. Lucas Numbers and Lucas Sequences
4.1. The Lucas Numbers
4.2. Criteria for Fibonacci Numbers
4.3. Prime Divisors of Fibonacci Numbers
4.4. Fibonacci Congruences
4.5. A More General Congruence
4.6. Narayana Sequence Congruences
4.7. Pythagorean Triples
4.8. Diophantine m-tuples
4.9. Generating Functions
4.10. Lucas Sequences
4.11. Pisano Period
4.12. π(n) and πL(n)
4.13. Prime Divisors of Lucas Numbers
5. Perfect Numbers and Fibonacci Primes
5.1. Square Sum Perfect Numbers
5.2. Some Lemmas
5.3. Proof of Theorems
5.4. A New Conjecture Concerning Perfect Numbers
5.5. Affine Square Sum Perfect Numbers
5.6. Square Sum Perfect Numbers and the Twin Prime Conjecture
5.7. Fermat Primes and GM Numbers
5.8. The abcd Equation
5.9. Applications of Elliptic Curves
5.10. Results with Lucas Sequences
Appendix
Appendix A.1. The First 100 Fibonacci Numbers and Their Prime Factorizations
Appendix A.2. The First 100 Lucas Numbers and Their Prime Factorizations
Bibliography
Index
