A Proof of Pillai’s Conjecture

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>1 and Y>1, for which with exponants strictly greater than 1, p>1 and q>1, Y^P=X^q+1 but for (X,Y,p,q) = (2,3,2,3) . We can verify that 3²=2³+1 Euler has proved that the equation Y²=X³+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 Y^P=X^q+a and prove that it has always a finite number of solutions for a fixed a.