\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}