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.
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