On the diophantine equation $x^p-x=y^q-y$

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On the diophantine equation $x^p-x=y^q-y$

Publicacions Matemàtiques


We consider the diophantine equation $$ x^p-x=y^q-y \tag"$(*)$" $$ in integers $(x,p,y,q)$. We prove that for given $p$ and $q$ with $2\le p < q$ $(*)$ has only finitely many solutions. Assuming the abc-conjecture we can prove that $p$ and $q$ are bounded. In the special case $p=2$ and $y$ a prime power we are able to solve $(*)$ completely.

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