About Pillai and Fermat-Catalan conjectures
Résumé
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 $X^3+1=Y^2$ has this only solution. We propose in this study a general solution. The particular cases already solved concern $p=2$, solved by Ko Chao in 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.
Domaines
Mathématiques générales [math.GM]
Origine : Fichiers produits par l'(les) auteur(s)