Generalized Cullen Numbers in Linear Recurrence Sequences

Abstract

A Cullen number is a number of the form $m{2}^m+1$, 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 $G_n=m{x}^m+T(x)$, 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 $T(x)$.