On the intersection of infinite geometric and arithmetic progressions

被引：0

作者：

Artūras Dubickas

Jonas Jankauskas

机构：

[1] Vilnius University,Department of Mathematics and Informatics

来源：

Bulletin of the Brazilian Mathematical Society, New Series | 2010年 / 41卷

关键词：

arithmetic and geometric progression; fractional part; algebraic number; linear recurrence; zero multiplicity; 11B25; 11B37; 11J71; 11R04; 11R06; 12D10;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

We prove that the intersection G ∩ A of an infinite geometric progression G = u, uq, uq2, uq3, ..., where u > 0 and q > 1 are real numbers, and an infinite arithmetic progression A contains at most 3 elements except for two kinds of ratios q. The first exception occurs for q = r1/d, where r > 1 is a rational number and d ∈ ℕ. Then this intersection can be of any cardinality s ∈ ℕ or infinite. The other (possible) exception may occur for q = β1/d, where β > 1 is a real cubic algebraic number with two nonreal conjugates of moduli distinct from β and d ∈ ℕ. In this (cubic) case, we prove that the intersection G ∩ A contains at most 6 elements.

引用

页码：551 / 566

页数：15

共 25 条

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