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:
PrefaceThe History of Perfect Numbers:
- What are Perfect Numbers?Euclid's ElementsNicomachusSums of Squares and CubesIbn al-HaythamMersenne Numbers and Mersenne PrimesFermat and DescartesThe Euclid-Euler TheoremIvan PervushinThe Lucas-Lehmer Primality TestThe Great Internet Mersenne Prime Search
Questions Related to Perfect Numbers: - Properties of Even Perfect NumbersOpen QuestionsOdd Perfect NumbersTouchard's TheoremDeficient and Abundant NumbersWeird Numbers and Semiperfect NumbersOre Numbers and the Harmonic MeanVariations on Ore NumbersAmicable NumbersMultiply Perfect NumbersThree Further GeneralizationsS -perfect NumbersThe Golden Ratio Conjecture
The Fibonacci Sequence: - Fibonacci, Leonardo of PisaThe Rabbit ProblemGeneral Terms and LimitsConnection with Continued FractionsThree IdentitiesEquations Between Binomial CoefficientsDivisibility SequencesZeckendorf's TheoremFrom Base 2 to Base 3Hilbert's Tenth Problem
Lucas Numbers and Lucas Sequences: - The Lucas NumbersCriteria for Fibonacci NumbersPrime Divisors of Fibonacci NumbersFibonacci CongruencesA More General CongruenceNarayana Sequence CongruencesPythagorean TriplesDiophantine m -tupplesGenerating FunctionsLucas SequencesPisano Periodπ
and πLPrime Divisors of Lucas Numbers
Perfect Numbers and Fibonacci Primes: - Square Sum Perfect NumbersSome LemmasProof of TheoremsA New Conjecture Concerning Perfect NumbersAffine Square Sum Perfect NumbersSquare Sum Perfect Numbers and the Twin Prime ConjectureFermat Primes and GM NumbersThe abcd EquationApplications of Elliptic CurvesResults with Lucas Sequences
Appendices: - The First 100 Fibonacci Numbers and Their Prime FactorizationsThe First 100 Lucas Numbers and Their Prime FactorizationsPrime Factorization of the First 100 Lucas Numbers
BibliographyIndex
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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