Jamel Ghanouchi, A Proof of Pillai’s Conjecture, BMSA Volume 1, Bulletin of Mathematical Sciences and Applications (Volume 1)
https://www.scipress.com/BMSA.1.52
Abstract:
    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>
Keywords:
    Algebraic Resolution, Catalan Equation, Diophantine Equations, Pillai Conjecture