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.