TY - JOUR
T1 - A Proof of Pillai’s Conjecture
AU - Ghanouchi, Jamel
JF - Bulletin of Mathematical Sciences and Applications
VL - 1
SP - 52
EP - 56
SN - 2278-9634
PY - 2012
PB - SciPress Ltd
DO - 10.18052/www.scipress.com/BMSA.1.52
UR - https://www.scipress.com/BMSA.1.52
KW - Algebraic Resolution
KW - Catalan Equation
KW - Diophantine Equations
KW - Pillai Conjecture
AB - 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, YP=Xq+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 YP=Xq+a and prove that it has always a finite number of solutions for a fixed a.
ER -