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 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.