Recurrence with prescribed number of residues

JOURNAL OF NUMBER THEORY

卷:215

Article

Dubickas, Arturas¹  Novikas, Aivaras¹ 

[1] Vilnius Univ, Fac Math & Informat, Inst Math, Naugarduko 24, LT-03225 Vilnius, Lithuania

关键词: Fibonacci numbers;    Lucas numbers;    Primitive divisors;    Fractional parts of powers;    Golden section;   

DOI  :  10.1016/j.jnt.2020.01.004

来源: Elsevier

【 摘 要 】

In this paper we show that for every positive integer m there exist positive integers x(1), x(2), M such that the sequence (x(n))(n=1)(infinity) defined by the Fibonacci recurrence x(n+2) = x(n+1) + x(n), n = 1, 2,3, ..., has exactly m distinct residues modulo M. As an application we show that for each integer m >= 2 there exists xi is an element of R such that the sequence of fractional parts {xi phi(n)}(n=)(infinity)(1), where phi = (1 + root 5)/2, has exactly m limit points. Furthermore, we prove that for no real xi not equal 0 it has exactly one limit point. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free