An Analytic Approach of some Conjectures Related to Diophantine Equations

Our purpose in this paper is to show how much diophantine equations are rich in analytic applications. Effectively, those equations allow to build sequences, series and numbers. The question of the analytic proof of some theorems remains of course, we will see it in this communication. We will make also an allusion to the Fermat numbers (x²ⁱ) and will see how this problem of the proof is actual and how it can be solved using the sequences and series.