\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 {amsfonts} \usepackage {latexsym} \usepackage {epsf} \setlength {\textwidth} {6.5in} \setlength {\oddsidemargin} {.1in} \setlength {\evensidemargin} {.1in} \setlength {\topmargin} {-.5in} \setlength {\textheight} {8.9in} \newcommand{\seqnum}[1] {\href{http://www.research.att.com/projects/OEIS?Anum=#1}{\underline{#1}}} \begin {document} \hyphenation {Fibonacci Lucas} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin {center} \vskip 1cm{\LARGE\bf Primes in Fibonacci $n$-step and\\ \vskip .1in Lucas $n$-step Sequences} \vskip 1cm \large Tony~D.~Noe\\ 14025~NW~Harvest~Lane\\ Portland,~OR~~97229\\ USA\\ \href{mailto:noe@sspectra.com}{\tt noe@sspectra.com} \\ \ \\ Jonathan~Vos~Post\\ 3225~North~Marengo~Avenue\\ Altadena,~CA~~91001\\ USA\\ \href{mailto:jvospost2@yahoo.com}{\tt jvospost2@yahoo.com} \end {center} \vskip .2in \begin {abstract} \noindent We search for primes in the Fibonacci $n$-step and Lucas $n$-step sequences, which are the natural generalizations of the Fibonacci and Lucas numbers. While the Fibonacci $n$-step sequences are nearly devoid of primes, the Lucas $n$-step sequences are prime-rich. We tabulate the occurrence of primes in the first 10000 terms for $n \leq 100$. We also state two conjectures about Diophantine equations based on these sequences. \end {abstract} \newtheorem {conjecture} {Conjecture} \section {Introduction} As illustrated in the tome by Koshy \cite {Koshy}, the Fibonacci and Lucas numbers are arguably two of the most interesting sequences in all of mathematics. The sequence of Fibonacci numbers $F_k$ is defined by the second-order linear recurrence formula and initial terms: \begin {equation*} F_{k+1}=F_k + F_{k-1}, \quad F_0=0, \quad F_1=1 \label {Eq.Fib} \end {equation*} Similarly, the sequence of Lucas numbers $L_k$ is defined by \begin {equation*} L_{k+1}=L_k + L_{k-1}, \quad L_0=2, \quad L_1=1 \label {Eq.Lucas} \end {equation*} Simple generalizations of these sequences are $F^{(n)}_k$, the Fibonacci $n$-step sequence, and $L^{(n)}_k$, the Lucas $n$-step sequence, which are defined by linear recurrence formulas of order $n>1$: \begin {eqnarray} F^{(n)}_{k+1}=F^{(n)}_k + F^{(n)}_{k-1} + \cdots + F^{(n)}_{k-n+1} \label {Eq.GenFib} \\ L^{(n)}_{k+1}=L^{(n)}_k + L^{(n)}_{k-1} + \cdots + L^{(n)}_{k-n+1} \label {Eq.GenLucas} \end {eqnarray} and $n$ initial terms \begin {alignat} {3} F^{(n)}_{1-n}&=1, &\quad F^{(n)}_k&=0, &\quad k&=-n+2, \ldots, 0 \label {Eq.InitFib} \\ L^{(n)}_0&=n, &\quad L^{(n)}_k&=-1, &\quad k&=-n+1, \ldots, -1 \label {Eq.InitLucas} \end {alignat} The sequence generated by equations \eqref {Eq.GenFib} and \eqref {Eq.InitFib} is also known as the $k$-generalized Fibonacci numbers, which are discussed by Flores \cite {Flores}. Observe that equations \eqref {Eq.GenFib} and \eqref {Eq.GenLucas} are equivalent to the three-term recursions \begin {eqnarray*} F^{(n)}_{k+1}=2F^{(n)}_k - F^{(n)}_{k-n} \label {Eq.GenFib2} \\ L^{(n)}_{k+1}=2L^{(n)}_k - L^{(n)}_{k-n} \label {Eq.GenLucas2} \end {eqnarray*} which are computationally superior for large $n$, and which show that all of these $n$-step sequences grow at a rate less than $2^k$. This recursion requires one more initial term, which we can take to be $F^{(n)}_1= L^{(n)}_1=1$. Also observe that the Lucas $n$-step sequence can be obtained from the Fibonacci $n$-step sequence with the identity \begin {equation*} L^{(n)}_{k} = F^{(n)}_k + 2 F^{(n)}_{k-1} + \cdots + (n-1) F^{(n)}_{k-n+2} + n F^{(n)}_{k-n+1} \\ \end {equation*} which follows from the generating functions of these two sequences. Dickson \cite {Dickson} cites a long history of generalizations of the Fibonacci numbers. Miles \cite {Miles} used equation \eqref {Eq.GenFib} in 1960. Fielder \cite {Fielder} appears to be the first to generalize the Lucas numbers to the Lucas $n$-step sequences. Catalani \cite {Catalani} has also written about these sequences. More recently, Benjamin and Quinn \cite {BQ} briefly discuss a combinatorial interpretation of these $n$-step sequences. The usual Fibonacci and Lucas numbers are obtained for $n=2$. For small values of $n$, these sequences are called tribonacci ($n=3$), tetranacci or quadranacci ($n=4$), pentanacci or pentacci ($n=5$), hexanacci or esanacci ($n=6$), heptanacci ($n=7$), and octanacci ($n=8$). Examples of these sequences and their A-numbers in Sloane's \cite {Sloane} OEIS database are given on the next page. \begin {table} [ht] \begin {center} \caption {Fibonacci $n$-step Sequences} \label {Ta.FibNum} {\fontsize 8 9 \selectfont \renewcommand {\arraystretch} {1.75} \begin {tabular} {| r r | c c c c c c c c c c c c c c | c |} \hline %& & & & & & & & & & & & & & & & \\ &$k$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & A-number \\ $n$ & & & & & & & & & & & & & & & & \\ \hline 2 & & 1 & 1 & \fbox {2} & \fbox {3} & \fbox {5} & 8 & \fbox {13} & 21 & 34 & 55 & \fbox {89} & 144 & \fbox {233} & 377 & \seqnum {A000045} \\ 3 & & 1 & 1 & 2 & 4 & \fbox {7} & \fbox {13} & 24 & 44 & 81 & \fbox {149} & 274 & 504 & 927 & 1705 & \seqnum {A000073} \\ 4 & & 1 & 1 & 2 & 4 & 8 & 15 & \fbox {29} & 56 & 108 & 208 & \fbox {401} & \fbox {773} & 1490 & 2872 & \seqnum {A000078} \\ 5 & & 1 & 1 & 2 & 4 & 8 & 16 & \fbox {31} & \fbox {61} & 120 & 236 & 464 & 912 & 1793 & 3525 & \seqnum {A001591} \\ 6 & & 1 & 1 & 2 & 4 & 8 & 16 & 32 & 63 & 125 & 248 & 492 & 976 & 1936 & 3840 & \seqnum {A001592} \\ 7 & & 1 & 1 & 2 & 4 & 8 & 16 & 32 & 64 & \fbox {127} & 253 & 504 & 1004 & 2000 & 3984 & \seqnum {A066178} \\ 8 & & 1 & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 255 & \fbox {509} & 1016 & 2028 & 4048 & \seqnum {A079262} \\ 9 & & 1 & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 511 & \fbox {1021} & 2040 & 4076 & \seqnum {A105753} \\ \hline \end {tabular} } \end {center} \end {table} \begin {table} [ht] \begin {center} \caption {Lucas $n$-step Sequences} \label {Ta.LucasNum} {\fontsize 8 9 \selectfont \renewcommand {\arraystretch} {1.75} \begin {tabular} {| r r | c c c c c c c c c c c c c | c |} \hline %& & & & & & & & & & & & & & & \\ &$k$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & A-number \\ $n$ & & & & & & & & & & & & & & & \\ \hline 2 & & 1 & \fbox {3} & 4 & \fbox {7} & \fbox {11} & 18 & \fbox {29} & \fbox {47} & 76 & 123 & \fbox {199} & 322 & \fbox {521} & \seqnum {A000032} \\ 3 & & 1 & 3 & 7 & 11 & 21 & 39 & \fbox {71} & \fbox {131} & \fbox {241} & \fbox {443} & 815 & \fbox {1499} & 2757 & \seqnum {A001644} \\ 4 & & 1 & 3 & 7 & 15 & 26 & 51 & 99 & \fbox {191} & \fbox {367} & 708 & 1365 & 2631 & 5071 & \seqnum {A073817} \\ 5 & & 1 & 3 & 7 & 15 & \fbox {31} & 57 & \fbox {113} & \fbox {223} & \fbox {439} & \fbox {863} & 1695 & 3333 & \fbox {6553} & \seqnum {A074048} \\ 6 & & 1 & 3 & 7 & 15 & 31 & 63 & 120 & \fbox {239} & 475 & 943 & \fbox {1871} & 3711 & 7359 & \seqnum {A074584} \\ 7 & & 1 & 3 & 7 & 15 & 31 & 63 & \fbox {127} & 247 & 493 & \fbox {983} & 1959 & 3903 & 7775 & \seqnum {A104621} \\ 8 & & 1 & 3 & 7 & 15 & 31 & 63 & 127 & 255 & 502 & 1003 & \fbox {2003} & 3999 & 7983 & \seqnum {A105754} \\ 9 & & 1 & 3 & 7 & 15 & 31 & 63 & 127 & 255 & 511 & \fbox {1013} & 2025 & 4047 & \fbox {8087} & \seqnum {A105755} \\ \hline \end {tabular} } \end {center} \end {table} The unique primes in each table appear in boxes. Note that these abbreviated sequences have many primes and that there appear to be more Lucas primes than Fibonacci primes. Also note that every Mersenne prime --- a prime of the form $2^p-1$ --- appears in these sequences as $F^{(p)}_{p+2}$ and $L^{(n)}_p$ for $n \geq p$. The primary purpose of this paper is to tabulate the values of $k$ that yield prime terms of the Fibonacci $n$-step and Lucas $n$-step sequences. Additionally, the paper describes how these primes were determined and makes conjectures about the distinct values in these sequences. \clearpage \section {Tables of Primes and Probable Primes} Tables \ref {Ta.Fib1} and \ref {Ta.Fib2}, at the end of this paper, list the $n$ and $k$ values that yield prime or probable prime (abbreviated \emph {prp}) terms of the Fibonacci $n$-step sequences $F^{(n)}_k$ for $2 \leq n \leq 100$ and $1 \leq k \leq 10000$. Similarly, Tables \ref {Ta.Lucas1}, \ref {Ta.Lucas2}, and \ref {Ta.Lucas3} list the $n$ and $k$ values that yield prime or probable prime terms of the Lucas $n$-step sequences $L^{(n)}_k$. A striking difference between the $F^{(n)}_k$ and $L^{(n)}_k$ tables is the small number of Fibonacci primes and the relatively large number of Lucas primes. This difference is explained by the parity of the two types of sequences. The fraction of odd numbers in the sequence $F^{(n)}_k$ is $2/(n+1)$ while the fraction of odd numbers in the sequence $L^{(n)}_k$ is either 1 or $n/(n+1)$, depending on whether $n$ is odd or even, respectively. Hence, because the $F^{(n)}_k$ sequence contains many fewer odd numbers, it contains many fewer prime numbers. In fact, using a probabilistic argument, such as in Hardy and Wright \cite [\S~2.5] {HW}, and taking into account the parity of the numbers in the sequences, we estimate the number of primes in $F^{(n)}_k$ for $k \leq N$: \begin {align*} \# \{ k \leq N: F^{(n)}_k \text {prime} \} &\thickapprox \frac {4}{n+1} \sum_{i=1}^{N} \frac {1}{\log F^{(n)}_i} \\ &\thickapprox \frac {4}{n+1} \sum_{i=1}^{N} \frac {1}{\log \alpha_n^i} = \frac {4}{n+1} \frac {H_N}{\log \alpha_n} \end {align*} where $\alpha_n$ is the growth rate of the sequence (the largest real root of $x^n (2-x)=1$) and \begin {equation*} H_N= \sum_{i=1}^{N} \frac {1}{i} \end {equation*} is the harmonic sum. Similarly, an estimate for the number of primes in $L^{(n)}_k$ for $k \leq N$ is \begin {equation*} \# \{ k \leq N: L^{(n)}_k \text {prime} \} \thickapprox \begin {cases} \frac {2 H_N}{\log \alpha_n}, & \text {if $n$ is odd};\\[6pt] \frac {n}{n+1} \frac {2H_N}{\log \alpha_n}, & \text {if $n$ is even.} \end {cases} \end {equation*} These estimates are evaluated for each $n$ and appear in the last column of the tables. In many cases, the estimates are close to the actual number of primes that we found. \section {Prime Search and Prime Proving} Each probable prime was determined using the PrimeQ function in \emph {Mathematica\textsuperscript{\textregistered}}. PrimeQ, which Wagon \cite {Wagon} discusses at length, is based on the Miller-Rabin strong pseudoprime test base 2 and base 3, and a Lucas test. Testing $F^{(n)}_k$ and $L^{(n)}_k$ with PrimeQ for $2 \leq n \leq 100$ and $1 \leq k \leq 10000$ required approximately 32 hours on a dual-processor 1.8 GHz G5 Macintosh\textsuperscript{\textregistered} computer. For small $n$, the search for probable primes has been more extensive than the limit $k \leq 10000$ used here. For instance, searches by Dubner and Keller \cite {DK99} and more recently by Lifchitz \cite {Lifchitz} have found probable prime values of $F^{(2)}_k$ and $L^{(2)}_k$ for $k>500000$. For the probable primes up to 1000 digits (and a few prps with more digits), we proved primality using an elliptic curve primality algorithm, as described by Atkin and Morain \cite {AM93}. This computation required approximately 60 hours on a 2.81 GHz Pentium\textsuperscript{\textregistered} 4 processor. All of the tested probable primes proved to be prime. For the Fibonacci and Lucas numbers, $F^{(2)}_k$ and $L^{(2)}_k$, Dubner and Keller \cite {DK99} have proved the primality for those entries in row $n=2$ of Tables \ref {Ta.Fib1} and \ref {Ta.Lucas1}. Probable primes in Tables \ref {Ta.Fib1}--\ref {Ta.Lucas3} are marked by an asterisk. \section {Duplicate Primes?} Are there duplicate primes in these tables? Let us ignore the initial terms of these $n$-step sequences because $m$-step and $n$-step sequences with $mm+1, \quad s>n+1 \end {equation*} has only two solutions: \begin {equation*} 13 = F^{(2)}_7 = F^{(3)}_6 \quad \text {and} \quad 504 = F^{(3)}_{12} = F^{(7)}_{11}. \end {equation*} \end {conjecture} \begin {conjecture} The Diophantine equation \begin {equation*} L^{(m)}_r = L^{(n)}_s,\quad r>m, \quad s>n \end {equation*} has only three solutions: \begin {equation*} 7 = L^{(2)}_4 = L^{(3)}_3, \quad 11 = L^{(2)}_{5} = L^{(3)}_4, \quad \text {and} \quad 5071 = L^{(3)}_{14} = L^{(4)}_{13}. \end {equation*} \end {conjecture} We have confirmed these conjectures for all terms whose magnitude is less than $2^{2000}$. Here we offer an heuristic argument showing that the expected number of solutions is small. Let $t$ be a positive integer and consider all the Fibonacci (or Lucas) $n$-step numbers (for any $n \leq t+1$) that are between $2^{t}$ and $2^{t+1}$. It is easy to see that any Fibonacci (or Lucas) $n$-step sequence has at most two numbers in this range for each $t$. Assuming that these numbers are randomly distributed in the range $2^{t}$ to $2^{t+1}$, the probability that at least two numbers are the same is $$1-\prod_{d=1}^{2t-1} (1-d/2^t)$$ Hence, an upper bound for the number of solutions is $$\sum_{t=1}^{\infty} \left( 1-\prod_{d=1}^{2t-1} (1-d/2^t)\right ) = 6.26505 \ldots$$ \section {Acknowledgment} The authors wish to acknowledge Eric Weisstein \cite {EWW} for inspiring this research. \begin {thebibliography} {99} \bibitem {AM93} A.~O.~L.~Atkin and F.~Morain, Elliptic Curves and Primality Proving, {\it Math. Comp.} \textbf {61} (1993), 29--68. \bibitem {BQ} A.~T.~Benjamin and J.~J.~Quinn, The Fibonacci numbers -- exposed more discretely, {\it Math. Mag.} \textbf {76} (2003), 182--192. \bibitem {Catalani} M.~Catalani, \htmladdnormallink {Polymatrix and generalized polynacci numbers} {http://www.arXiv.org/math/0210201}, published electronically at www.arXiv.org/math/0210201 \bibitem {Dickson} L.~E.~Dickson, \emph {History of the Theory of Numbers}, Vol.~1, AMS Chelsea, 1919. \bibitem {DK99} H.~Dubner and W.~Keller, New Fibonacci and Lucas primes, {\it Math. Comp.} \textbf {68} (1999), 417--427. \bibitem {Fielder} D.~C.~Fielder, Certain Lucas-like sequences and their generation by partitions of numbers, {\it Fibonacci Quart.}, \textbf {5} (1967), 319 \bibitem {Flores} I.~Flores, Direct calculation of k-generalized Fibonacci numbers, {\it Fibonacci Quart.}, \textbf {5} (1967), 259--266. \bibitem {HW} G.~H.~Hardy and E.~M.~Wright, \emph {An Introduction to the Theory of Numbers}, fourth ed., Oxford University Press, 1960. \bibitem {Koshy} T.~Koshy, \emph {Fibonacci and Lucas Numbers with Applications}, John Wiley and Sons, NY, 2001. \bibitem {Lifchitz} H.~Lifchitz, {\em \htmladdnormallink {Probable primes top 10000} {http://www.primenumbers.net/prptop/prptop.php}}, published electronically at \\ www.primenumbers.net/prptop/prptop.php \bibitem {Miles} E.~P.~Miles,~Jr., Generalized Fibonacci numbers and associated matrices, {\it Amer. Math. Monthly} \textbf {67} (1960), 745--752. \bibitem {Sloane} N.~J.~A.~Sloane, {\em \htmladdnormallink {The On-Line Encyclopedia of Integer Sequences}{http://www.research.att.com/~njas/sequences}}, published electronically at www.research.att.com/$\sim$njas/sequences. \bibitem {Wagon} S.~Wagon, \emph {Mathematica\textsuperscript{\textregistered} in Action}, second ed., Springer-Telos, New York, 2000. \bibitem {EWW} E.~W.~Weisstein, {\em \htmladdnormallink {Fibonacci n-step number} {http://mathworld.wolfram.com/Fibonaccin-StepNumber.html}}, published electronically at\\ mathworld.wolfram.com/Fibonaccin-StepNumber.html \end {thebibliography} \begin {table} \begin {center} \caption {Primes of the form \(F^{(n)}_k\)} \label {Ta.Fib1} {\fontsize 8 9 \selectfont \begin {tabular} {|r|p {4in}|c|c|} \hline & & number of & est. \# of\\ $n$ & \quad $k$ for which $F^{(n)}_k$ is prime or prp(*) & primes for & primes for\\[3pt] & & \(k \leq 10000\) & \(k \leq 10000\) \\ \hline 2 & 3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677 \seqnum {A001605} \seqnum {A005478} & 26 & 27 \\ 3 & 3, 5, 6, 10, 86, 97, 214, 801, 4201 \seqnum {A092835} \seqnum {A092836} & 9 & 16 \\ 4 & 3, 7, 11, 12, 36, 56, 401, 2707, 8417* \seqnum {A104534} \seqnum {A104535} & 9 & 12 \\ 5 & 3, 7, 8, 25, 146, 169, 182, 751, 812, 1507, 1591, 3157, 3752 \seqnum {A105756} \seqnum {A105757} & 13 & 10 \\ 6 & 3, 36, 37, 92, 660, 6091*, 8415* \seqnum {A105758} \seqnum {A105759} & 7 & 8 \\ 7 & 3, 9, 17 \seqnum {A105760} \seqnum {A105761} & 3 & 7 \\ 8 & 3, 11, 19, 119, 344, 1316 & 6 & 6 \\ 9 & 3, 12, 871 & 3 & 6 \\ 10 & 3, 353, 365, 2432 & 4 & 5 \\ 11 & 3, 14, 50, 325, 2510 & 5 & 5 \\ 12 & 3, 80, 1237 & 3 & 4 \\ 13 & 3, 15, 16, 8051* & 4 & 4 \\ 14 & 3, 361, 1471 & 3 & 4 \\ 15 & 3, 34, 82, 657, 5361*, 7026* & 6 & 4 \\ 16 & 3, 274, 4098, 4506 & 4 & 3 \\ 17 & 3, 19, 37, 1459 & 4 & 3 \\ 18 & 3, 59, 3079 & 3 & 3 \\ 19 & 3, 21, 22, 2982, 4382* & 5 & 3 \\ 20 & 3, 233, 484 & 3 & 3 \\ 21 & 3, 24, 46, 4072* & 4 & 3 \\ 22 & 3, 47, 232, 393 & 4 & 2 \\ 23 & 3, 26 & 2 & 2 \\ 24 & 3, 52, 451 & 3 & 2 \\ 25 & 3, 287 & 2 & 2 \\ 26 & 3, 487 & 2 & 2 \\ 27 & 3 & 1 & 2 \\ 28 & 3, 31 & 2 & 2 \\ 29 & 3 & 1 & 2 \\ 30 & 3, 1118 & 2 & 2 \\ 31 & 3, 33, 7233* & 3 & 2 \\ 32 & 3, 167 & 2 & 2 \\ 33 & 3 & 1 & 2 \\ 34 & 3, 9942 & 2 & 2 \\ 35 & 3, 254, 3745, 6013* & 4 & 2 \\ 36 & 3 & 1 & 2 \\ 37 & 3, 723 & 2 & 1 \\ 38 & 3 & 1 & 1 \\ 39 & 3 & 1 & 1 \\ 40 & 3, 247 & 2 & 1 \\ 41 & 3 & 1 & 1 \\ 42 & 3, 518, 1205 & 3 & 1 \\ 43 & 3 & 1 & 1 \\ 44 & 3 & 1 & 1 \\ 45 & 3 & 1 & 1 \\ 46 & 3 & 1 & 1 \\ 47 & 3, 194, 865, 2401 & 4 & 1 \\ 48 & 3, 6322* & 2 & 1 \\ 49 & 3 & 1 & 1 \\ 50 & 3 & 1 & 1 \\ \hline \end {tabular} } \end {center} \end {table} \begin {table} \begin {center} \caption {Primes of the form \(F^{(n)}_k\), continued} \label {Ta.Fib2} {\fontsize 8 9 \selectfont \begin {tabular} {|r|p {4in}|c|c|} \hline & & number of & est. \# of\\ $n$ & \quad $k$ for which $F^{(n)}_k$ is prime or prp(*) & primes for & primes for\\[3pt] & & \(k \leq 10000\) & \(k \leq 10000\) \\ \hline 51 & 3 & 1 & 1 \\ 52 & 3 & 1 & 1 \\ 53 & 3, 2809, 4052*, 4592* & 4 & 1 \\ 54 & 3, 2366 & 2 & 1 \\ 55 & 3 & 1 & 1 \\ 56 & 3 & 1 & 1 \\ 57 & 3 & 1 & 1 \\ 58 & 3, 120, 3718 & 3 & 1 \\ 59 & 3 & 1 & 1 \\ 60 & 3, 1405 & 2 & 1 \\ 61 & 3, 63, 374 & 3 & 1 \\ 62 & 3, 8695* & 2 & 1 \\ 63 & 3, 5057* & 2 & 1 \\ 64 & 3 & 1 & 1 \\ 65 & 3, 925 & 2 & 1 \\ 66 & 3, 269 & 2 & 1 \\ 67 & 3, 341 & 2 & 1 \\ 68 & 3 & 1 & 1 \\ 69 & 3 & 1 & 1 \\ 70 & 3, 1564 & 2 & 1 \\ 71 & 3 & 1 & 1 \\ 72 & 3 & 1 & 1 \\ 73 & 3 & 1 & 1 \\ 74 & 3, 302, 2476 & 3 & 1 \\ 75 & 3 & 1 & 1 \\ 76 & 3 & 1 & 1 \\ 77 & 3, 4916* & 2 & 1 \\ 78 & 3 & 1 & 1 \\ 79 & 3 & 1 & 1 \\ 80 & 3 & 1 & 1 \\ 81 & 3 & 1 & 1 \\ 82 & 3 & 1 & 1 \\ 83 & 3 & 1 & 1 \\ 84 & 3 & 1 & 1 \\ 85 & 3 & 1 & 1 \\ 86 & 3, 263, 8701* & 3 & 1 \\ 87 & 3 & 1 & 1 \\ 88 & 3 & 1 & 1 \\ 89 & 3, 91, 811 & 3 & 1 \\ 90 & 3 & 1 & 1 \\ 91 & 3 & 1 & 1 \\ 92 & 3, 1024 & 2 & 1 \\ 93 & 3, 96 & 2 & 1 \\ 94 & 3 & 1 & 1 \\ 95 & 3 & 1 & 1 \\ 96 & 3 & 1 & 1 \\ 97 & 3 & 1 & 1 \\ 98 & 3, 299 & 2 & 1 \\ 99 & 3 & 1 & 1 \\ 100 & 3 & 1 & 1 \\ \hline \end {tabular} } \end {center} \end {table} \begin {table} \begin {center} \caption {Primes of the form \(L^{(n)}_k\)} \label {Ta.Lucas1} {\fontsize 8 9 \selectfont \begin {tabular} {|r|p {4in}|c|c|} \hline & & number of & est. \# of\\ $n$ & \quad $k$ for which $L^{(n)}_k$ is prime or prp(*) & primes for & primes for\\[3pt] & & \(k \leq 10000\) & \(k \leq 10000\) \\ \hline 2 & 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467 \seqnum {A001606} \seqnum {A005479} & 32 & 27 \\ 3 & 2, 3, 4, 7, 8, 9, 10, 12, 20, 30, 33, 66, 76, 77, 82, 87, 98, 180, 205, 360, 553, 719, 766, 1390, 1879, 1999, 4033*, 5620* \seqnum {A104576} \seqnum {A105762} & 28 & 32 \\ 4 & 2, 3, 8, 9, 16, 19, 24, 27, 46, 68, 71, 78, 107, 198, 309, 377, 477, 1057, 1631, 2419, 3974*, 4293*, 8247* \seqnum {A104577} \seqnum {A105763} & 23 & 24 \\ 5 & 2, 3, 5, 7, 8, 9, 10, 13, 30, 35, 77, 98, 126, 160, 192, 810, 1086, 1999, 2021, 3157, 3426*, 3471* \seqnum {A105764} \seqnum {A105765} & 22 & 29 \\ 6 & 2, 3, 5, 8, 11, 32, 37, 46, 123, 237, 332, 408, 772, 827, 1523, 5610* \seqnum {A105766} \seqnum {A105767} & 16 & 24 \\ 7 & 2, 3, 5, 7, 10, 17, 24, 25, 26, 28, 38, 40, 49, 62, 79, 89, 114, 140, 145, 182, 248, 353, 437, 654, 702, 784, 921, 931, 986, 1206, 2136, 2137, 3351*, 5411* \seqnum {A104622} \seqnum {A105768} & 34 & 28 \\ 8 & 2, 3, 5, 7, 11, 16, 17, 112, 140, 159, 186, 347, 425, 565, 2643, 2931, 3314, 4767*, 9015* & 19 & 25 \\ 9 & 2, 3, 5, 7, 10, 13, 23, 27, 38, 41, 56, 84, 107, 112, 123, 203, 209, 758, 765, 895, 1830, 3172, 6763*, 7709* & 24 & 28 \\ 10 & 2, 3, 5, 7, 48, 59, 75, 79, 156, 167, 245, 335, 370, 412, 970, 1358, 1479, 1652 & 18 & 26 \\ 11 & 2, 3, 5, 7, 15, 31, 32, 39, 41, 47, 66, 89, 111, 173, 182, 295, 334, 350, 357, 480, 548, 662, 681, 853, 1100, 1128, 1171, 1328, 2203, 2427, 2817, 2891, 6818* & 33 & 28 \\ 12 & 2, 3, 5, 7, 15, 16, 46, 69, 112, 120, 144, 411, 574, 1009, 1140, 1191, 1257, 1358, 2809, 3223, 3512*, 3759*, 4441*, 6099*, 8304*, 9673*, 9734* & 27 & 26 \\ 13 & 2, 3, 5, 7, 13, 14, 22, 27, 53, 55, 92, 242, 299, 344, 382, 699, 767, 834, 1057, 1270, 1565, 1797, 1837, 3847*, 4328*, 4911*, 5093*, 6664*, 7313*, 7470*, 9789* & 31 & 28 \\ 14 & 2, 3, 5, 7, 13, 22, 23, 52, 76, 82, 119, 125, 243, 281, 400, 437, 659, 994, 1377, 1426, 1940, 2303, 3853*, 6947* & 24 & 26 \\ 15 & 2, 3, 5, 7, 13, 16, 31, 37, 38, 47, 48, 58, 66, 132, 167, 173, 269, 375, 542, 612, 911, 1073, 2821, 3237, 3656*, 3910*, 4424*, 4432*, 6978* & 29 & 28 \\ 16 & 2, 3, 5, 7, 13, 26, 131, 145, 243, 821, 863, 1388, 1481, 4457*, 5309*, 8696* & 16 & 27 \\ 17 & 2, 3, 5, 7, 13, 17, 59, 208, 325, 353, 926, 991, 1201, 1230, 1347, 1764, 8101* & 17 & 28 \\ 18 & 2, 3, 5, 7, 13, 17, 23, 33, 45, 58, 107, 134, 217, 286, 552, 985, 1193, 2455, 4202*, 6176*, 8710*, 9831* & 22 & 27 \\ 19 & 2, 3, 5, 7, 13, 17, 19, 91, 208, 219, 306, 312, 439, 494, 640, 656, 702, 872, 3346*, 3874*, 7769*, 7804* & 22 & 28 \\ 20 & 2, 3, 5, 7, 13, 17, 19, 27, 35, 47, 53, 76, 88, 243, 281, 288, 355, 406, 436, 541, 761, 1055, 2343, 2585, 3705*, 4965*, 7033* & 27 & 27 \\ 21 & 2, 3, 5, 7, 13, 17, 19, 28, 52, 61, 165, 263, 278, 284, 691, 1110, 1130, 1372, 1526, 2542, 2656, 3403*, 4162*, 5645*, 6156*, 6572*, 8496* & 27 & 28 \\ 22 & 2, 3, 5, 7, 13, 17, 19, 34, 113, 191, 288, 451, 561, 646, 867, 1842, 2368, 5369*, 5827*, 5993* & 20 & 27 \\ 23 & 2, 3, 5, 7, 13, 17, 19, 28, 37, 44, 86, 133, 142, 164, 307, 700, 2250, 3255, 9785* & 19 & 28 \\ 24 & 2, 3, 5, 7, 13, 17, 19, 70, 79, 87, 99, 104, 140, 153, 214, 556, 743, 766, 905, 1824, 3983*, 4807* & 22 & 27 \\ 25 & 2, 3, 5, 7, 13, 17, 19, 26, 39, 43, 46, 49, 65, 70, 79, 98, 188, 218, 337, 452, 539, 904, 3185, 5530* & 24 & 28 \\ 26 & 2, 3, 5, 7, 13, 17, 19, 28, 35, 38, 61, 161, 263, 374, 475, 653, 668, 816, 914, 1073, 1841, 2353, 2794, 3176, 4278*, 4845*, 5305*, 6803*, 7748*, 8618* & 30 & 27 \\ 27 & 2, 3, 5, 7, 13, 17, 19, 31, 52, 58, 101, 112, 209, 244, 281, 288, 326, 449, 468, 585, 718, 4814*, 5478*, 7622* & 24 & 28 \\ 28 & 2, 3, 5, 7, 13, 17, 19, 35, 68, 86, 88, 89, 98, 102, 141, 160, 689, 785, 840, 1140, 1730, 1978, 2187, 3315 & 24 & 27 \\ 29 & 2, 3, 5, 7, 13, 17, 19, 44, 73, 85, 132, 135, 214, 221, 259, 358, 411, 426, 502, 740, 987, 1033, 1594, 1626, 2530, 2551, 2669, 5090*, 7568*, 9821* & 30 & 28 \\ 30 & 2, 3, 5, 7, 13, 17, 19, 32, 41, 44, 45, 75, 82, 99, 107, 292, 338, 371, 1108, 2983, 4783*, 7932* & 22 & 27 \\ 31 & 2, 3, 5, 7, 13, 17, 19, 31, 43, 45, 50, 52, 57, 81, 137, 172, 291, 472, 497, 502, 1003, 1746, 2007, 2115, 2226, 2318, 6116*, 7580* & 28 & 28 \\ 32 & 2, 3, 5, 7, 13, 17, 19, 31, 35, 51, 57, 59, 65, 73, 85, 106, 124, 303, 318, 481, 503, 507, 681, 1624, 2633, 4186*, 4945* & 27 & 27 \\ 33 & 2, 3, 5, 7, 13, 17, 19, 31, 44, 49, 52, 70, 71, 82, 85, 129, 138, 236, 249, 327, 704, 1373, 1406, 1477, 3370*, 3430* & 26 & 28 \\ \hline \end {tabular} } \end {center} \end {table} \begin {table} \begin {center} \caption {Primes of the form \(L^{(n)}_k\), continued} \label {Ta.Lucas2} {\fontsize 8 9 \selectfont \begin {tabular} {|r|p {4in}|c|c|} \hline & & number of & est. \# of\\ $n$ & \quad $k$ for which $L^{(n)}_k$ is prime or prp(*) & primes for & primes for\\[3pt] & & \(k \leq 10000\) & \(k \leq 10000\) \\ \hline 34 & 2, 3, 5, 7, 13, 17, 19, 31, 73, 85, 104, 135, 244, 296, 317, 360, 366, 373, 397, 470, 1238, 2118, 2125, 2594, 5870*, 8742* & 26 & 27 \\ 35 & 2, 3, 5, 7, 13, 17, 19, 31, 38, 50, 91, 97, 109, 132, 276, 361, 535, 1281, 1720, 2285, 2302, 3971*, 4322*, 5177*, 7336* & 25 & 28 \\ 36 & 2, 3, 5, 7, 13, 17, 19, 31, 53, 117, 230, 298, 367, 701, 797, 846, 1008, 2122, 2210 & 19 & 27 \\ 37 & 2, 3, 5, 7, 13, 17, 19, 31, 69, 101, 341, 349, 577, 1147, 1963, 4542*, 8999* & 17 & 28 \\ 38 & 2, 3, 5, 7, 13, 17, 19, 31, 44, 50, 58, 70, 77, 89, 145, 181, 197, 211, 247, 460, 475, 498, 579, 625, 687, 1288, 1736, 1740, 4172*, 7810*, 9629* & 31 & 28 \\ 39 & 2, 3, 5, 7, 13, 17, 19, 31, 52, 85, 91, 243, 867, 1106, 1167, 1633, 2684, 2917, 3733*, 4593*, 5055* & 21 & 28 \\ 40 & 2, 3, 5, 7, 13, 17, 19, 31, 47, 56, 64, 70, 154, 230, 714, 995, 2478, 3965*, 6837*, 8498* & 20 & 28 \\ 41 & 2, 3, 5, 7, 13, 17, 19, 31, 45, 81, 91, 109, 131, 236, 405, 616, 803, 2886, 3875*, 5014*, 7930*, 9405* & 22 & 28 \\ 42 & 2, 3, 5, 7, 13, 17, 19, 31, 53, 77, 95, 111, 119, 429, 463, 729, 879, 904, 1182, 3467*, 5562*, 6347* & 22 & 28 \\ 43 & 2, 3, 5, 7, 13, 17, 19, 31, 50, 75, 81, 99, 192, 315, 338, 362, 369, 499, 1202, 1926, 2304, 6049* & 22 & 28 \\ 44 & 2, 3, 5, 7, 13, 17, 19, 31, 93, 99, 179, 227, 337, 385, 550, 613, 795, 2506, 2627, 3910*, 6169* & 21 & 28 \\ 45 & 2, 3, 5, 7, 13, 17, 19, 31, 87, 127, 176, 219, 295, 331, 420, 498, 697, 814, 2174, 2535, 6243*, 7158*, 8429*, 9065* & 24 & 28 \\ 46 & 2, 3, 5, 7, 13, 17, 19, 31, 53, 65, 76, 93, 98, 154, 181, 194, 595, 644, 1829, 1994, 2410, 3242, 7321* & 23 & 28 \\ 47 & 2, 3, 5, 7, 13, 17, 19, 31, 55, 56, 97, 128, 185, 192, 304, 348, 405, 543, 631, 750, 942, 1492, 1889, 2276, 4861*, 6206*, 6882*, 7001*, 7764*, 8146* & 30 & 28 \\ 48 & 2, 3, 5, 7, 13, 17, 19, 31, 59, 69, 92, 123, 135, 283, 308, 387, 726, 897, 1325, 3561*, 4130*, 5153*, 9109* & 23 & 28 \\ 49 & 2, 3, 5, 7, 13, 17, 19, 31, 50, 52, 87, 92, 129, 131, 185, 287, 494, 706, 728, 793, 1271, 1297, 1655, 1954, 2965, 3674*, 5129*, 5823*, 8847*, 9667* & 30 & 28 \\ 50 & 2, 3, 5, 7, 13, 17, 19, 31, 57, 99, 107, 116, 161, 290, 518, 685, 688, 1044, 7182* & 19 & 28 \\ 51 & 2, 3, 5, 7, 13, 17, 19, 31, 74, 97, 112, 184, 380, 382, 397, 411, 646, 3092, 6313*, 7128*, 7917* & 21 & 28 \\ 52 & 2, 3, 5, 7, 13, 17, 19, 31, 57, 62, 76, 95, 192, 223, 333, 752, 1518, 2479, 3980*, 6056*, 7620* & 21 & 28 \\ 53 & 2, 3, 5, 7, 13, 17, 19, 31, 79, 115, 116, 128, 129, 141, 243, 348, 380, 503, 664, 1135, 1212, 1250, 2221, 2417, 3141, 3841*, 4322*, 5007*, 6038*, 6161*, 6521*, 7344*, 7385* & 33 & 28 \\ 54 & 2, 3, 5, 7, 13, 17, 19, 31, 59, 76, 82, 223, 285, 485, 791, 941, 1215, 1224, 1339, 3649*, 3737*, 5594*, 5758*, 5907* & 24 & 28 \\ 55 & 2, 3, 5, 7, 13, 17, 19, 31, 56, 112, 142, 309, 499, 607, 1109, 1648, 2866, 3170, 4006*, 4345*, 5494*, 7059* & 22 & 28 \\ 56 & 2, 3, 5, 7, 13, 17, 19, 31, 116, 178, 204, 303, 390, 475, 725, 1190, 2031, 5112*, 5234*, 8130*, 9776* & 21 & 28 \\ 57 & 2, 3, 5, 7, 13, 17, 19, 31, 62, 69, 70, 187, 201, 425, 596, 928, 1125, 1951, 6164*, 7609* & 20 & 28 \\ 58 & 2, 3, 5, 7, 13, 17, 19, 31, 82, 83, 92, 94, 112, 515, 793, 1117, 1385, 1471, 3503*, 6102*, 8559* & 21 & 28 \\ 59 & 2, 3, 5, 7, 13, 17, 19, 31, 64, 68, 2260, 8086*, 9189* & 13 & 28 \\ 60 & 2, 3, 5, 7, 13, 17, 19, 31, 99, 113, 172, 238, 576, 1877, 5904*, 8860* & 16 & 28 \\ 61 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 64, 68, 165, 291, 480, 1140, 1214, 1371, 2384, 2717, 3773*, 4203*, 5534*, 6924* & 23 & 28 \\ 62 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 68, 74, 131, 287, 593, 798, 4705*, 4827*, 5384* & 18 & 28 \\ 63 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 70, 92, 116, 135, 145, 152, 377, 387, 437, 603, 690, 695, 803, 1215, 1217, 1367, 1781, 2347, 2406, 7764*, 9297* & 30 & 28 \\ 64 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 77, 113, 163, 265, 351, 369, 824, 906, 2589, 2731, 3516*, 4820*, 9498* & 22 & 28 \\ 65 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 81, 93, 175, 276, 307, 382, 804, 1279, 1668, 2067, 2189, 3235, 5452* & 22 & 28 \\ 66 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 76, 196, 340, 382, 391, 655, 1276, 1289, 1419, 1488, 2314, 2795, 7465*, 8961* & 23 & 28 \\ 67 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 87, 116, 117, 207, 221, 226, 391, 402, 578, 733, 961, 984, 1078, 1107, 2019, 3232, 6755*, 6799*, 7198*, 9892* & 29 & 28 \\ \hline \end {tabular} } \end {center} \end {table} \begin {table} \begin {center} \caption {Primes of the form \(L^{(n)}_k\), continued} \label {Ta.Lucas3} {\fontsize 8 9 \selectfont \begin {tabular} {|r|p {4in}|c|c|} \hline & & number of & est. \# of\\ $n$ & \quad $k$ for which $L^{(n)}_k$ is prime or prp(*) & primes for & primes for\\[3pt] & & \(k \leq 10000\) & \(k \leq 10000\) \\ \hline 68 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 107, 131, 139, 241, 369, 388, 435, 577, 782, 813, 1605, 1730, 1770, 5429*, 5482*, 7001*, 7277*, 7635*, 8019* & 28 & 28 \\ 69 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 70, 81, 94, 116, 146, 167, 176, 185, 237, 463, 563, 722, 1501, 1953, 3049, 5437* & 25 & 28 \\ 70 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 98, 123, 124, 187, 285, 582, 613, 761, 826, 1127, 1199, 2723, 3966*, 5867*, 6575*, 7888* & 25 & 28 \\ 71 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 103, 115, 129, 187, 238, 425, 1349, 1688, 2354, 2554, 5197*, 8616* & 21 & 28 \\ 72 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 113, 386, 393, 443, 811, 1045, 1229, 2032, 3721*, 5478*, 5935*, 6610*, 7391*, 7406*, 7664*, 7769*, 9063* & 26 & 28 \\ 73 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 87, 93, 242, 401, 409, 495, 604, 772, 893, 1682, 4864*, 5852* & 21 & 28 \\ 74 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 93, 99, 101, 199, 452, 823, 1934, 2994 & 17 & 28 \\ 75 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 81, 123, 166, 213, 370, 515, 763, 2210, 2512, 2840, 3177, 8727*, 9893* & 22 & 28 \\ 76 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 128, 146, 261, 304, 491, 1005, 1032, 1229, 1671, 1887, 2835, 3723*, 3836*, 9200* & 23 & 28 \\ 77 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 133, 142, 154, 182, 187, 206, 444, 604, 796, 839, 916, 1152, 1528, 1541, 2222, 3002, 3662*, 3768*, 5792*, 7384*, 9444*, 9495* & 31 & 28 \\ 78 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 88, 368, 555, 719, 1101, 1418, 1904, 4475*, 4676*, 5287*, 5585* & 20 & 28 \\ 79 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 81, 91, 161, 251, 476, 535, 590, 662, 814, 841, 971, 1097, 1205, 2967, 3052, 5403*, 6036*, 7671* & 27 & 28 \\ 80 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 87, 93, 249, 366, 461, 528, 535, 632, 675, 844, 1157, 1894, 2068, 2241, 2849, 3617*, 6028*, 7381*, 8614* & 28 & 28 \\ 81 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 99, 141, 233, 235, 290, 425, 535, 615, 649, 1317, 1722, 5129*, 5831*, 8044*, 8595*, 8681* & 25 & 28 \\ 82 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 161, 286, 287, 788, 2402, 6383*, 7236*, 8738* & 17 & 28 \\ 83 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 249, 422, 423, 1031, 1062, 1180, 2637, 2836, 4919*, 5641*, 8043*, 8197* & 21 & 28 \\ 84 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 95, 219, 303, 736, 1120, 1447, 1590, 2076, 2788, 4163*, 4490* & 20 & 28 \\ 85 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 127, 221, 740, 1607, 3139, 3800*, 5533*, 8133*, 9747* & 18 & 28 \\ 86 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 93, 299, 379, 416, 470, 606, 662, 1791, 2908, 3999* & 19 & 28 \\ 87 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 109, 121, 269, 326, 367, 551, 593, 883, 1083, 3036, 3294, 5433*, 8169*, 9523* & 23 & 28 \\ 88 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 107, 112, 129, 493, 1173, 1549, 3407*, 3424*, 5895* & 18 & 28 \\ 89 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 164, 169, 1829, 1880, 2888, 3408*, 3694*, 5173*, 5567*, 7612*, 7983* & 21 & 28 \\ 90 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 202, 256, 262, 297, 373, 383, 485, 844, 3390*, 5261*, 5533*, 6743*, 7572* & 23 & 28 \\ 91 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 169, 201, 250, 526, 570, 731, 807, 964, 1062, 2515, 2908, 3139 & 22 & 28 \\ 92 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 164, 170, 196, 215, 225, 284, 353, 375, 379, 491, 628, 668, 1148, 1888, 2135, 4069*, 6331*, 6483*, 7904* & 29 & 28 \\ 93 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 112, 146, 225, 470, 568, 599, 654, 941, 993, 3012, 7306* & 21 & 28 \\ 94 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 159, 325, 581, 708, 743, 1260, 1527, 1983, 4575*, 6507*, 7747* & 21 & 28 \\ 95 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 176, 181, 193, 219, 283, 348, 387, 392, 527, 549, 761, 766, 848, 852, 911, 1041, 1067, 3721*, 3914*, 4304*, 5552*, 5905*, 7409* & 33 & 28 \\ 96 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 125, 166, 184, 287, 316, 392, 822, 1252, 2095, 2185, 3056, 3235, 3663*, 7381*, 8634* & 25 & 28 \\ 97 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 104, 213, 274, 697, 1982, 2908, 3678*, 4236* & 18 & 28 \\ 98 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 161, 221, 478, 591, 748, 825, 860, 943, 1488, 1567, 4615*, 5292*, 8338* & 23 & 28 \\ 99 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 129, 261, 348, 639, 1301, 1531, 4151*, 6775* & 18 & 28 \\ 100 & 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 124, 179, 211, 243, 531, 599, 814, 934, 1529, 3803*, 4768*, 5496*, 6554*, 7858*, 8764*, 9083*, 9976* & 27 & 28 \\ \hline \end {tabular} } \end {center} \end {table} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11A41; Secondary 11B39. \noindent \emph {Keywords:} prime numbers, generalized Fibonacci and Lucas numbers. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum {A000032}, %2-Lucas \seqnum {A000045}, %2-Fibonacci \seqnum {A000073}, %3-Fibonacci \seqnum {A000078}, %4-Fibonacci \seqnum {A001591}, %5-Fibonacci \seqnum {A001592}, %6-Fibonacci \seqnum {A001605}, %2-Fibonacci prime n \seqnum {A001606}, %2-Lucas prime n \seqnum {A001644}, %3-Lucas \seqnum {A005478}, %2-Fibonacci prime \seqnum {A005479}, %2-Lucas prime \seqnum {A066178}, %7-Fibonacci \seqnum {A073817}, %4-Lucas \seqnum {A074048}, %5-Lucas \seqnum {A074584}, %6-Lucas \seqnum {A079262}, %8-Fibonacci \seqnum {A092835}, %3-Fibonacci prime n \seqnum {A092836}, %3-Fibonacci prime \seqnum {A104534}, %4-Fibonacci prime n \seqnum {A104535}, %4-Fibonacci prime \seqnum {A104576}, %3-Lucas prime n \seqnum {A104577}, %4-Lucas prime n \seqnum {A104621}, %7-Lucas \seqnum {A104622}, %7-Lucas prime n \seqnum {A105753}, %9-Fibonacci \seqnum {A105754}, %8-Lucas \seqnum {A105755}, %9-Lucas \seqnum {A105756}, %5-Fibonacci prime n \seqnum {A105757}, %5-Fibonacci prime \seqnum {A105758}, %6-Fibonacci prime n \seqnum {A105759}, %6-Fibonacci prime \seqnum {A105760}, %7-Fibonacci prime n \seqnum {A105761}, %7-Fibonacci prime \seqnum {A105762}, %3-Lucas prime \seqnum {A105763}, %4-Lucas prime \seqnum {A105764}, %5-Lucas prime n \seqnum {A105765}, %5-Lucas prime \seqnum {A105766}, %6-Lucas prime n \seqnum {A105767}, %6-Lucas prime and \seqnum {A105768}.) %7-Lucas prime \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 19 2005; revised version received September 5 2005. Published in {\it Journal of Integer Sequences}, September 20 2005. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}