Generalized Cullen numbers in linear recurrence sequences

JOURNAL OF NUMBER THEORY

卷:202

Article

Bilu, Yuri¹  Marques, Diego²  Togbe, Alain³ 

[1] Univ Bordeaux 1, IMB, 351 Cours Liberat, F-33405 Talence, France

[2] Univ Brasilia, Dept Matemat, BR-70910900 Brasilia, DF, Brazil

[3] Purdue Univ Northwest, Dept Math Stat & Comp Sci, 1401 S,US 421, Westville, IN 46391 USA

关键词: Cullen numbers;    Linear forms in logarithms;    Linear recurrence sequence;    Diophantine equations;   

DOI  :  10.1016/j.jnt.2018.11.025

来源: Elsevier

【 摘 要 】

A Cullen number is a number of the form m2(m) +1, where m is a positive integer. In 2004; Luca and Stanica proved, among other things, that the largest Fibonacci number in the Cullen sequence is F-4 = 3. Actually, they searched for generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence (G(n))(n), under weak assumptions, and a given polynomial T(x) is an element of Z[x], we shall prove that if G(n) = mx(m) + T(x), then m << 1 and n << log vertical bar x vertical bar, where the implied constants depend only on (G(n))(n), and T(x). (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free