On the diophantine equation $x^p-x=y^q-y$
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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.
$$
x^p-x=y^q-y
\tag"$(*)$"
$$
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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Com citar
Mignotte, M.; Pethö, Attila , 1950-. «On the diophantine equation $x^p-x=y^q-y$». Publicacions Matemàtiques, 1999, vol.VOL 43, núm. 1, p. 207-16, https://raco.cat/index.php/PublicacionsMatematiques/article/view/37959.