On the diophantine equation x[p] - x = y[q] - y
On the diophantine equation x[p] - x = y[q] - y
复制标题
关于丢番图方程 x[p] - x = y[q] - y
DOI：
10.5565/37959
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发表时间：
1999
影响因子：
1.1
通讯作者：
Pethö Attila
中科院分区：
文献类型：
--
作者：
M. Mignotte;Pethö Attila
我们考虑丢番图方程
$$
x^p-x=y^q-y
\tag”$（*）$”
$$
整数$（x，p，y，q）$。我们证明了对于给定的p和q，其中p < q（*）只有2个解。假设abc猜想成立，我们可以证明p和q是有界的。在特殊情况下$p=2$和$y$是一个素数幂，我们能够完全解决$（*）$。
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.