About Catalan-Mihailescu Theorem

(MSC = 11) more than one century ago, the Belgian mathematician Eugene Catalan has formulated a famous conjecture. It became a theorem in 2004. The theorem stipulates that the following equation Yp = 1+Xq has only one solution, which is 32 = 1+23 when X>1, Y>1, p>1, q>1 all integers. We prove in this research firstly that Catalan equation is equivalent to the following equation Yq-p = Xp-1. After a little change of the data of the problem, we prove also that Catalan equation implies two other equations. Those equations allow to define convergent sequences. It is the Algebraic-Analytic approach which conducts to the impossibility of Catalan equation for p>2. The equation is simplified to the case p = 2, q = 3. It becomes consequently easy to prove that the only solution of Catalan equation is (X,Y,p,q) = (2,3,2,3).