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 3^{2}=2^{3}+1 Euler has proved that the equation Y^{2}=X^{3}+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.*

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 1)

Pages:

52-56

DOI:

10.18052/www.scipress.com/BMSA.1.52

Citation:

J. Ghanouchi "A Proof of Pillai’s Conjecture", Bulletin of Mathematical Sciences and Applications, Vol. 1, pp. 52-56, 2012

Online since:

Aug 2012

Authors:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License