As a teacher of several mathematics subjects at university level, and writer of several books that preceded to this one, and as a result of my goal to try to explain what seemed difficult as something easy, for a few years I began to investigate in several mathematics areas about possible simple proofs to complex mathematical problems. This book contains the results of these investigations, referring to Fermat's last theorem, as well as the existence of solutions for the Fermat equation in other fields such as quadratic integers and Gaussians, and conjectures such as Collatz conjecture and Goldbach strong conjecture.