论文标题
有效绑定了广义的二聚体m-tupels
An effective bound on Generalized Diophantine m-tuples
论文作者
论文摘要
For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $S = \{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j \ leq m $。定义$ m_k（n）：= \ sup \ {| s | ：s $具有属性$ d_k（n）\} $。在最近的一项工作中，第二作者S. Kim和M. R. Murty证明了$ m_k（n）$是$ o（\ log n）$，对于固定的$ k $，因为我们不同$ n $。在本文中，我们获得了$ m_k（n）$的有效上限。特别是，我们表明，对于$ k \ geq 2 $，$ m_k（n）\ leq 3 \，ϕ（k）\，\ log n $，如果$ n $足够大于$ k $。
For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $S = \{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j\leq m$. Define $M_k(n):= \sup\{|S| : S$ has property $D_k(n)\}$. In a recent work, the second author, S. Kim and M. R. Murty proved that $M_k(n)$ is $O(\log n)$, for a fixed $k$, as we vary $n$. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$, $M_k(n) \leq 3\,ϕ(k)\, \log n$, if $n$ is sufficiently larger than $k$.
