On the diophantine equation x[p] - x = y[q] - y

Autores: A. Pethö, Maurice Mignotte
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 43, Nº 1, 1999, págs. 207-216
Idioma: inglés
DOI: 10.5565/publmat_43199_08
Títulos paralelos:
- Sobre la ecuación diofántica xp - x = yq - y
Enlaces
- Texto completo
Resumen
- 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.