\documentclass[12pt,reqno]{article}
\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}
\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}
\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}
\usepackage{color}
\usepackage{fullpage}
\usepackage{float}
\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.1in}
\setlength{\textheight}{8.4in}
\newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}}
\begin{document}
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\begin{center}
\vskip 1cm{\LARGE\bf
Fibonacci $s$-Cullen and $s$-Woodall Numbers
}
\vskip 1cm
\large
Diego Marques\\
Departamento de Matem\' atica\\
Universidade de Bras\' ilia\\
Bras\' ilia, Brazil\\
\href{mailto:diego@mat.unb.br}{\tt diego@mat.unb.br}\\
\ \\
Ana Paula Chaves\\
Instituto de Matem\' atica e Estat\' istica\\
Universidade Federal de Goi\' as\\
Goi\' as, Brazil\\
\href{mailto:apchaves@ufg.br}{\tt apchaves@ufg.br}\\
\end{center}
\vskip .2 in
\newcommand{\R}{{\mathbb R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\N}{{\mathbb N}}
\hyphenation{non-ze-ro}
\hyphenation{in-vi-si-ble}
\renewcommand{\baselinestretch}{1}
\def\noi {\noindent}
\def\Ker {{\rm Ker}}
\def\Im {{\rm Im}}
\def\a {\alpha}
\def\B {\mathcal{B}}
\def\O {\mathcal{O}}
\def\N {\mathbb{N}}
\def\Z {\mathbb{Z}}
\def\Q {\mathbb{Q}}
\def\R {\mathbb{R}}
\def\RR {\mathcal{R}}
\def\QQ {\overline{\Q}}
\def\QQQ {\QQ}
\def\C {\mathbb{C}}
\def\cF {\mathcal{F}}
\def\cR {\mathcal{R}}
\def\e {\epsilon}
\def\d {\mathfrak{d}}
\def\g {\gamma}
\def\fle {\longrightarrow }
\def\k {\kappa}
\begin{abstract}
The $m$-th Cullen number $C_m$ is a number of the form $m2^m+1$ and the $m$-th Woodall number $W_m$ has the form $m2^m-1$. In 2003, Luca and St\u anic\u a proved that the largest Fibonacci number in the Cullen sequence is $F_4=3$ and that $F_1=F_2=1$ are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given $s>1$, the equation $F_n=ms^m\pm 1$ has only finitely many solutions, and they are effectively
computable.
In this note, we shall provide the explicit form of the possible solutions.
\end{abstract}
\section{Introduction}\label{sec:1}
A {\it Cullen number} is a number of the form $m2^m+1$ (denoted by $C_m$), where $m$ is a nonnegative integer. This sequence was introduced in 1905 by Father J. Cullen \cite{cul} and it was mentioned in the well-known book of Guy \cite[Section {\bf B20}]{guy}. These numbers gained great interest in 1976, when
Hooley \cite{hoo} showed that almost all Cullen numbers are composite. However, despite being very scarce, it is still conjectured that there are infinitely many \textit{Cullen primes}.
In a similar way, a \textit{Woodall number} (also called \textit{Cullen
number of the second kind}) is a positive integer of the form $m2^m-1$
(denoted by $W_m$). It is also known that almost all Woodall numbers are
composite. However, it is also conjectured that the set of {\it
Woodall primes} is infinite.
These numbers can be generalized to the \textit{$s$-Cullen and $s$-Woodall numbers} which are numbers of the form
\begin{center}
$C_{m,s}=ms^m+1$ and $W_{m,s}=ms^m-1$,
\end{center}
where $m\geq 1$ and $s\geq 2$. This family was introduced by
Dubner \cite{dub}. A prime of the form $C_{m,s}$ is $C_{139948,151}$ an integer with $304949$ digits.
Many authors have searched for special properties of Cullen and Woodall numbers and their generalizations. We refer the reader to \cite{pt,uber,mdc,pseudo} for classical and recent results on this subject.
In 2003, Luca and St\u anic\u a \cite[Theorem 3]{LS} proved that the largest Fibonacci number in the Cullen sequence is $F_4=3=1\cdot 2^1+1$ and that $F_1=F_2=1=1\cdot 2^1-1$ are the largest Fibonacci numbers in the Woodall sequence.
Recall that $\nu_p(r)$ denotes the $p$-adic order of $r$,
which is the exponent of the highest power of a prime $p$ which divides $r$. Also, the {\it order (\mbox{or} rank) of appearance} of $n$ in the Fibonacci sequence, denoted by $z(n)$, is defined as the smallest positive integer $k$, such that $n\mid F_k$ (for results on this function, see \cite{d20} and references therein). Let $p$ be a prime number and set $e(p):=\nu_p(F_{z(p)})$.
Very recently, Marques \cite{JIS} proved that if the equation
\begin{equation}\label{Main}
F_n=ms^{m}+\ell
\end{equation}
has solution, with $m>1$ and $\ell\in \{\pm 1\}$, then $m<(6.2 + 1.9e(p))\log (3.1+e(p))$, for some prime factor $p$ of $s$. This together with the fact that $e(p)=1$ for all prime $p<2.8\cdot 10^{16}$ (PrimeGrid, March 2014) implies that there is no Fibonacci number that is also a nontrivial (i.e., $m>1$) $s$-Cullen number or $s$-Woodall number when the set of prime divisors of $s$ is contained in $\{2,3,5,\ldots, 27999999999999991\}$. This is the set of the first $759997990476073$ prime numbers.
In particular, the previous result ensures that for any given $s\geq 2$, there are only finitely many Fibonacci numbers which are also $s$-Cullen numbers or $s$-Woodall numbers and they are effectively computable.
In this note, we shall invoke the primitive divisor theorem to provide explicitly the possible values of $m$ satisfying Eq.~(\ref{Main}). More precisely,
\begin{theorem}\label{main1}
Let $s>1$ be an integer. Let $(n,m,\ell)$ be a solution of the Diophantine equation (\ref{Main}) with $n, m>1$ and $\ell\in \{-1,1\}$. Then $m=e(p)/\nu_p(s)$, for some prime factor $p$ of $s$.
\end{theorem}
In particular, we have that $m\leq e(p)$ for some prime factor $p$ of
$s$. Also, we can deduce \cite[Corollary 3]{JIS} from the above
theorem. In fact, for all $p<2.8\cdot 10^{16}$ we have $e(p)=1$ and
then if $(n,m,\ell)$ is a solution, with $m>1$, we would have the
contradiction that $11$.
So we may suppose that $n\geq 28$. We rewrite Eq.~(\ref{Main}) as
$F_n-\ell=ms^{m}$. It is well-known that $F_n\pm 1=F_aL_b$, where
$2a,2b\in \{n\pm 2,n\pm 1\}$. (This factorization depends on the class
of $n$ modulo $4$. See \cite[(3)]{PM} for more details.) Then the main
equation becomes
\[
F_aL_b=ms^{m},
\]
where $2a,2b\in \{n\pm 2,n\pm 1\}$ and $|a-b|\in \{1,2\}$. Since $a-b\in \{\pm 1, \pm 2\}$, then $\gcd(a,b)\in \{1,2\}$ and then $\gcd(F_a,L_b)=1, 2$ or $3$. Therefore, we have $F_a=m_1s_1^m$ and $L_b=m_2s_2^m$, where $m_1m_2=m, s_1s_2=s$ and $\gcd(m_1,m_2),\gcd(s_1,s_2)\in \{1,2,3\}$. We claim that $s_1>1$. Suppose, to get a contradiction, that $s_1=1$, then $F_a=m_1$ and $L_b=m_2s^m$. Since $2a-4\geq n-6\geq (n+8)/2\geq b+3$, we arrive at the following contradiction:
\[
m^2\geq m_1^2=F_a^2\geq \a^{2a-4}\geq \a^{b+3}\geq 2L_b=2m_2s^m\geq 2^{m+1}>m^2,
\]
where $\a=(1+\sqrt{5})/2$. Here, we used that $F_{j}\geq \a^{j-2}$ and $L_j\leq \a^{j+1}$. Thus $s_1>1$. Since $a\geq (n-2)/2\geq 13$, then by the primitive divisor theorem (see \cite{char}), there exists a primitive divisor $p$ of $F_a$ (i.e., $p\mid F_a$ and $p\nmid F_1\cdots F_{a-1}$). We also have that $p\equiv \pm 1\pmod a$. In particular, $p\geq a-1$. Thus $p\mid F_a=m_1s_1^m$. Suppose that $p\mid m_1$. In this case, one has that $a-1\leq p\leq m_1\leq m$. On the other hand, we get
\[
2^m\leq m_1s_1^m=F_a\leq \alpha^{a-1}<2^{a-1}.
\]
Thus $m3, s=s_1s_2$ and $\gcd(s_1,s_2)\leq 3$). On the other hand, $z(p)=a$ and so $\nu_p(F_{z(p)})=\nu_p(F_a)=m\nu_p(s)$ as desired.
\qed
\section{Acknowledgements}
The first author is grateful to FAP-DF and CNPq for financial support.
The authors wish to thank the editor and the referee for their helpful
comments.
\begin{thebibliography}{99}
\bibitem{char} R. D. Carmichael, On the numerical factors of the arithmetic forms $\alpha^n\pm \beta^n$. \textit{Ann. of Math.} \textbf{15} (1913), 30--70.
\bibitem{cul} J. Cullen, Question 15897. \textit{Educ. Times},
\textbf{534} (1905).
\bibitem{dub} H. Dubner, Generalized Cullen numbers. \textit{J.
Recreat. Math.}, \textbf{21} (1989), 190--194.
\bibitem{pt} J. M. Grau and A. M. Oller-Marc\' en, An
$\tilde{O}(\log^2(N))$ time primality test for generalized Cullen
numbers. \textit{Math. Comp.} {\bf 80} (2011), 2315--2323.
\bibitem{guy} R. Guy, \textit{Unsolved Problems in Number Theory}. 2nd ed.,
Springer-Verlag, 1994.
\bibitem{uber} F. Heppner, $\ddot{\mbox{U}}$ber Primzahlen der Form
$n2^n+ 1$ bzw. $p2^p+ 1$. \textit{Monatsh. Math.} \textbf{85} (1978),
99--103.
\bibitem{hoo} C. Hooley, \textit{Applications of the Sieve Methods to
the Theory of Numbers}. Cambridge University Press, Cambridge, 1976.
\bibitem{LS} F. Luca and P. St\u anic\u a, Cullen numbers in
binary recurrent sequences, in {\it Applications of Fibonacci Numbers},
vol.~10, Kluwer Academic Publishers, 2004, pp.~167--175.
\bibitem{mdc} F. Luca, On the greatest common divisor of two Cullen
numbers. \textit{Abh. Math. Sem. Univ. Hamburg} \textbf{73} (2003),
253--270.
\bibitem{pseudo} F. Luca and I. Shparlinski, Pseudoprime Cullen and
Woodall numbers. \textit{Colloq. Math.} \textbf{107} (2007), 35--43.
\bibitem{JIS} D. Marques, On generalized Cullen and Woodall numbers
which are also Fibonacci numbers. \textit{J. Integer Sequences}, {\bf
17} (2014),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL17/Marques/marques5r2.html}{Article 14.9.4}.
\bibitem{PM} D. Marques, The Fibonacci version of the Brocard-Ramanujan
Diophantine equation. \textit{Port. Math.} \textbf{68} (2011),
185--189.
\bibitem{d20} D. Marques, Sharper upper bounds for the order of
appearance in the Fibonacci sequence. \textit{Fibonacci Quart.}
\textbf{50} (2013), 233--238.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39.
\noindent \emph{Keywords: }
Fibonacci number, Cullen number.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000045} and
\seqnum{A002064}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received October 11 2014;
revised version received December 24 2015.
Published in {\it Journal of Integer Sequences}, January 6 2015.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}