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.