Pré-Publication, Document De Travail Année : 2012

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.

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hal-00751215 , version 1 (13-11-2012)
hal-00751215 , version 2 (13-11-2012)
hal-00751215 , version 3 (17-11-2012)
hal-00751215 , version 4 (22-11-2012)
hal-00751215 , version 5 (25-11-2012)
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hal-00751215 , version 7 (27-11-2012)
hal-00751215 , version 8 (28-11-2012)
hal-00751215 , version 9 (31-01-2013)
hal-00751215 , version 10 (31-01-2013)
hal-00751215 , version 11 (28-03-2013)
hal-00751215 , version 12 (13-03-2014)

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Jamel Ghannouchi. About Pillai and Fermat-Catalan conjectures. 2012. ⟨hal-00751215v1⟩
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