An Algebraic-Analytic Approach of Diophantine Equations

Our purpose in this research is to show how much Fermat equation is rich in analytic applications. Effectively, this equation allows to build amazing sequences, series and numbers. The question of the elementary proof of the theorem remains of course, we will see it in this communication. We will make also an allusion tothe very known Fermat numbers We will see how this problem of the proof is actual and how it can be solved using Fermat sequences and series.