发表机构
Universidade de Brasília（巴西利亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了埃尔德什-马勒猜想的邻分母情形，即若无穷多n对应的P(pnqnqn+1)有界，则无理数ξ为刘维尔数，证明结合了连分数行列式恒等式与p进对数线性型估计。
AI 中文摘要
1939年，埃尔德什和马勒猜想：对于无理实数ξ，若无穷多个连分数收敛项pn/qn的P(pnqn)有界，则ξ必为刘维尔数，其中P(N)表示非零整数N的最大素因子。本文证明了该猜想的邻分母情形：若无穷多个n对应的P(pnqnqn+1)有界，则ξ为刘维尔数。证明结合了连分数收敛项的行列式恒等式与p进对数线性型的固定基估计。
英文摘要
We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $ξ$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(S)>0$ such that \[ \log q_{n+1}\gg_{ξ,S} q_n^c \] along those indices. Consequently, $ξ$ is a Liouville number. The proof uses the determinant identity for consecutive convergents and a fixed-base consequence of Yu's theorem on $p$-adic logarithmic forms.
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