Jamel Ghanouchi, A Proof of Pillai’s Conjecture, BMSA Volume 1, Bulletin of Mathematical Sciences and Applications (Volume 1)
    Catalan theorem has been proved in 2002 by Preda Mihailescu. In 2004, it became officially Catalan-Mihailescu theorem. This theorem stipulates that there are not consecutive pure powers. There do not exist integers stricly greater than 1, X&gt;1 and Y&gt;1, for which with exponants strictly greater than 1, p&gt;1 and q&gt;1, Y<sup>P</sup>=X<sup>q</sup>+1 but for (X,Y,p,q) = (2,3,2,3) . We can verify that 3<sup>2</sup>=2<sup>3</sup>+1 Euler has proved that the equation Y<sup>2</sup>=X<sup>3</sup>+1 has this only solution. We propose in this study a general solution. The particular cases already solved concern p=2, solved by Ko Chaoin 1965, and q=3 which has been solved in 2002. The case q=2 has been solved by Lebesgue in 1850. We solve here the equation for the general case. We generalize the proof to Pillai’s conjecture <i>Y<sup>P</sup>=X<sup>q</sup>+a </i> and prove that it has always a finite number of solutions for a fixed <i>a.</i>
    Algebraic Resolution, Catalan Equation, Diophantine Equations, Pillai Conjecture