PERFECT NUMBERS AND FIBONACCI SEQUENCES

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 SearchQuestions 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 ConjectureThe Fibonacci Sequence:Fibonacci, Leonardo of PisaThe Rabbit ProblemGeneral Terms and LimitsConnection with Continued Fractions Three IdentitiesEquations Between Binomial CoefficientsDivisibility SequencesZeckendorf's TheoremFrom Base 2 to Base 3Hilbert's Tenth ProblemLucas 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(n) and πL(n)Prime Divisors of Lucas NumbersPerfect 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 SequencesAppendices:The First 100 Fibonacci Numbers and Their Prime FactorizationsThe First 100 Lucas Numbers and Their Prime FactorizationsPrime Factorization of the First 100 Lucas NumbersBibliographyIndex
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.

Key Features:
The competitive advantage is described as above, moreover, as Albert Einstein said after he discovered the theory of relativity, "the real law is nonlinear"The discovery of square perfect number and its necessary and sufficient condition confirms this