发表机构
National Institute of Technology Calicut（印度国家技术学院卡利卡特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究完美幂的稀疏表示，证明了一类含 $S$-单位系数的方程解有限，并扩展了 Bennett-Bugeaud 结果，指出固定底数下五数字完美幂不存在。
AI 中文摘要
本文研究了关于 $S$-单位方程中的完美幂以及完美幂的稀疏表示的两个问题。Corvaja-Zannier 证明了在二进制展开中恰好具有四个非零数字的奇数完美幂在自然数集中仅有有限多个。首先，我们证明了方程 $y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}$（其中 $0<m_1<m_2<m_3$，且 $c_i$ 属于某个合适集合 $S$ 的 $S$-单位）的解集的有限性。该证明结合了子空间定理关于解析级数在 $S$-单位点处整数值的推论与数域上的 Roth 定理。作为第二个结果，我们扩展了 Bennett 和 Bugeaud 的工作，证明了：对于每个固定的整数底数 $g\geq 2$，存在常数 $q_0(g)$，使得对于每个素数 $q>q_0(g)$，不存在在底数 $g$ 下具有五个非零数字表示的完美 $q$ 次幂。该证明依赖于 Archimedean 与非 Archimedean 对数中线性形式的显式下界。
英文摘要
In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\N$ having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation \begin{equation*} y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}, \quad 0<m_1<m_2<m_3, \end{equation*} where $c_i$ are $S$-units for a suitable set $S$. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at $S$-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base $g\geq 2$, there exists a constant $q_0(g)$ such that, for every prime $q>q_0(g)$, no perfect $q$-th power admits a representation with five non-zero digits in base $g$. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.
Comments18 pages
