More than one century after its formulation by the Belgian mathematician Eugene Catalan, Preda Mihailescu has solved the open problem. But, is it all ? Mihailescu's solution utilizes computation on machines, we propose here not really a proof of Catalan theorem as it is entended classically, but a resolution of an equation like the resolution of the polynomial equations of third and fourth degrees. This solution is totally algebraic and does not utilize, of course, computers or any kind of calculation. We generalize our approach to Beal equation and discuss the solutions. (Keywords: Diophantine equations, Catalan, Fermat-Catalan, Conjectures, Proofs, Algebraic resolution).
Bulletin of Mathematical Sciences and Applications (Volume 5)