Generalized Cullen Numbers in Linear Recurrence Sequences

TL;DR

This paper investigates generalized Cullen numbers within higher order linear recurrence sequences, establishing bounds on their parameters under certain conditions, extending previous results from Fibonacci to more complex recurrences.

Contribution

It extends the study of Cullen numbers to higher order linear recurrence sequences and provides explicit bounds on their parameters, generalizing prior work on Fibonacci sequences.

Findings

Bounds on m in terms of log log |x| and log^2(log log |x|)

Bounds on n in terms of log |x| and log log |x|

Results depend only on the recurrence sequence and polynomial T(x)

Abstract

A Cullen number is a number of the form $m2^m+1$, where $m$ is a positive integer. In 2004, Luca and St\u anic\u a 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 $T(x)\in \mathbb{Z}[x]$, we shall prove that if $G_n=mx^m+T(x)$, then \[ m\ll\log \log |x|\log^2(\log \log |x|)\ \mbox{and}\ n\ll\log |x|\log\log |x|\log^2(\log \log |x|), \] where the implied constant depends only on $(G_n)_n$ and $T(x)$.