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\begin{center}
\vskip 1cm{\LARGE\bf 
A Tribonacci-Like Sequence of  \\
\vskip .1in
Composite Numbers
}
\vskip 1cm
\large
Ivan Lunev \\ 
Department of Mathematics and Mechanics \\ 
Saint-Petersburg State University \\
28 Universitetsky Prospect \\
St.~Petersburg,  198504  \\
Russia \\
\href{mailto: lun-vano@yandex.ru}{\tt lun-vano@yandex.ru}
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\begin{abstract}
We find a new Tribonacci-like sequence of positive integers $\langle x_0,x_1,x_2,\ldots\rangle$ given by $x_n=x_{n-1}+x_{n-2}+x_{n-3}$ ,  $n\ge 3$, and $\gcd(x_0,x_1,x_2)=1$ that contains no prime numbers. We show that the sequence with initial values $x_0=151646890045$, $x_1=836564809606$, $x_2=942785024683$ is the current record in terms of the number of digits.
\end{abstract}

\section{Introduction}
\v{S}iurys \cite{Siurys} found initial values 
\begin{align*}
x_0 &amp;=99202581681909167232 \\
x_1 &amp;=67600144946390082339 \\
x_2 &amp;=139344212815127987596,
\end{align*}
satisfying $\gcd(x_0,x_1,x_2)=1$,
such that the Tribonacci-like sequence given by 
\begin{align} \label{a1}
 x_n=x_{n-1}+x_{n-2}+x_{n-3} \text{ for } n\ge 3
\end{align}
contains no prime numbers.
Similar problems were considered for Fibonacci-like sequences given by
$x_n=x_{n-1}+x_{n-2}$ for $n\ge 2$ (Graham \cite{Graham}; Knuth
\cite{Knuth}; Wilf \cite{Wilf}; Nicol \cite{Nicol}; Vsemirnov
\cite{Vsemirnov}; Ismailescu and Son \cite{Ismailescu}),
sequences given by $a_n=k 2^n+1$ (Sierpi\'{n}ski \cite{Sierpinski};
Jaeschke \cite{Jaeschke}), binary linear recurrent sequences (
Dubickas, Novikas, and \v{S}iurys \cite{Dubickas}; Somer
\cite{Somer}) and some linear recurrent sequences of higher orders 
(\v{S}iurys \cite{Siurys1}).

The main result of this note is as follows.

\section{The main results}
\begin{theorem}
 \label{Thm:MainResult}
Let $\langle x_0,x_1,x_2,\ldots\rangle$ be defined by~\eqref{a1} and $\gcd(x_0,x_1,x_2)=1$ with the following initial values:
\begin{equation*} 
x_0=151646890045,\quad
x_1=836564809606,\quad
x_2=942785024683.
\end{equation*}
	Then $\langle x_0,x_1,x_2,\ldots\rangle$ contains no prime numbers.
\end{theorem}

\begin{remark}
  \label{rem:Rem}
If we allow non-positive values, we can find a slightly smaller (in absolute value) initial triple, namely 
\begin{equation*}
x_0= 730344594529,\quad
x_1= -45426674968,\quad
x_2= 151646890045.
\end{equation*}
\end{remark}

\section{Proof of Theorem~\ref{Thm:MainResult}}
In this section we complete the proof of Theorem~\ref{Thm:MainResult}.

\begin{proof}[Proof of Theorem~\ref{Thm:MainResult}]
First, recall \v{S}iurys' idea \cite{Siurys}.
Consider the additional sequences $(s_n)_{n=0}^\infty$ and $(t_n)_{n=0}^\infty$ defined by the same relation~\eqref{a1} with  $(s_0, s_1, s_2)=(0,1,0)$ and $(t_0, t_1, t_2) = (0,0,1)$.

\begin{lemma}[\cite{Siurys}]
  \label{lem:Technical}
Let $p$ be a prime. Suppose that for some integer $m\ge 2$ we have $s_mt_{2m}-s_{2m}t_m\equiv 0 \pmod{p}$. Then there exist $a$, $b\in\mathbb{Z}$ such that at least one of $a$, $b$ is not divisible by $p$ and  $s_{km}a+t_{km}b\equiv 0 \pmod{p}$ for $k=0,1,2,\ldots$.
\end{lemma}

The next step is to find a set of pairs $(p_i, m_i)$ satisfying Lemma \ref{lem:Technical} such that every integer belongs to at least one of the arithmetic progressions

\begin{align} \label{a2}
   A_i = m_ik + r_i, k \in \mathbb{Z} , i = 1,2,\ldots ,11 .
\end{align}
	
In this paper, the following values of $p_i$ and $m_i$ are used: (see Table \ref{tab:InformativeTable}).

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\mathbf{i}$ &amp; $\mathbf{p_i}$ &amp; $\mathbf{m_i}$ &amp; $\mathbf{|s_{m_i}t_{2m_i}-s_{2m_i}t_{m_i}|}$ \\
\hline
$1$ &amp; $2$ &amp; $2$ &amp; $\textbf2$ \\ 
\hline
$2$ &amp; $29$ &amp; $5$ &amp; $\textbf{29}$ \\
\hline
$3$ &amp; $17$ &amp; $6$ &amp; $2\cdot\textbf{17}$ \\
\hline
$4$ &amp; $7$ &amp; $8$ &amp; $2^6\cdot\textbf7$ \\
\hline
$5$ &amp; $11$ &amp; $10$ &amp; $2\cdot\textbf{11}\cdot29$ \\
\hline
$6$ &amp; $107$ &amp; $12$ &amp; $2^3\cdot17\cdot\textbf{107}$ \\
\hline
$7$ &amp; $8819$ &amp; $15$ &amp; $29\cdot\textbf{8819}$ \\
\hline
$8$ &amp; $19$ &amp; $20$ &amp; $2^3\cdot11\cdot\textbf{19}\cdot29\cdot\textbf{239}$ \\
\hline
$9$ &amp; $239$ &amp; $20$ &amp; $2^3\cdot11\cdot\textbf{19}\cdot29\cdot\textbf{239}$ \\
\hline
$10$ &amp; $1151$ &amp; $24$ &amp; $2^6\cdot7\cdot17\cdot107\cdot\textbf{1151}$ \\
\hline
$11$ &amp; $1621$ &amp; $30$ &amp; $2\cdot11\cdot17\cdot29\cdot\textbf{1621}\cdot8819$ \\
\hline
\end{tabular}
\end{center}
  \caption{\label{tab:InformativeTable} $p_i$ and $m_i$.}
\end{table}

\v{S}iurys \cite{Siurys} used $p = 79$ with $m = 40$ instead of $p = 239$.

By Lemma \ref{lem:Technical}, for every pair $(p_i, m_i)$ we can choose $(a_i,b_i)\in \mathbb{Z}^2$ so that at least one of $a_i$, $b_i$ is not divisible by $p_i$ and

\begin{equation*}
   s_{km_i}a_i+t_{km_i}b_i\equiv 0 \pmod{p_i} \text{ for } k=0,1,2,\ldots .
\end{equation*}
Next, we construct a sequence $(x_n)_{n=0}^\infty$  satisfying
	
\begin{equation*}
x_n\equiv s_{m_i-r_i+n}a_i+t_{m_i-r_i+n}b_i \pmod{p_i},\qquad  i=1,2,\ldots ,11,\qquad \text{ for } n=0,1,2,\ldots .
\end{equation*}
The initial values satisfy
	
$$
\begin{aligned}
&amp; x_0\equiv s_{m_i-r_i}a_i+t_{m_i-r_i}b_i  \pmod{p_i}, \qquad x_1\equiv s_{m_i-r_i+1}a_i+t_{m_i-r_i+1}b_i \pmod{p_i},\\
&amp; x_2\equiv s_{m_i-r_i+2}a_i+t_{m_i-r_i+2}b_i \pmod{p_i},\qquad  \text{ for } i=1,2,\ldots ,11.	\\
\end{aligned}
$$

We can find initial terms $(x_0,x_1,x_2)$ by the Chinese reminder theorem.

In the method described above there is some freedom in the choice of $a_i$ and $b_i$ (up to a common factor). \v{S}iurys \cite{Siurys} used all $a_i$ equal to $1$.

We show how to optimize the choice of $a_i$ and $b_i$. Let $P=\prod_{i=1}^{11}p_i$ .

Let us consider the system: 
\begin{equation*}
\begin{cases}
x_0'\equiv Dx_0 \pmod{P}\\
x_1'\equiv Dx_1 \pmod{P}\\
x_2'\equiv Dx_2 \pmod{P}
\end{cases}
\end{equation*}
subject to the constraint
\begin{equation} \label{a3}
    \gcd(D,P)=1.
\end{equation}
	
The new triple $(x_0', x_1', x_2')$ also satisfies the above properties, 
i.e., each term of the sequence~\eqref{a1} with starting values $(x_0', x_1', x_2')$ is 
divisible by at least one of $p_1,\ldots,p_{11}$.
	
For a moment, let us forget about condition~\eqref{a3}. Then the problem can 
be formulated as follows: find the minimum vector of the form

\begin{equation*}
D(x_0,x_1,x_2)+U_1(P,0,0)+U_2(0,P,0)+U_3(0,0,P),
\end{equation*}
i.e., the vector in the lattice generated by the vectors $(x_0,x_1,x_2)$, $(P,0,0)$, $(0,P,0)$, $(0,0,P)$. The smallest vector can be found by the LLL-algorithm \cite{Lenstra}.

For any admissible covering~\eqref{a2} with the above $m_i$'s we build the initial values $(x_0,x_1,x_2)$ for the sequence~\eqref{a1}
by using \v{S}iurys' method. Then using the LLL-algorithm we find the smallest lattice basis. Coordinates $(x_0', x_1', x_2')$ for each of the new three basis vectors will suit us, if the condition~\eqref{a3} is satisfied and $(x_0', x_1', x_2')$ are of the same sign 
(if all of them are negative, then replace $(x_0', x_1', x_2')$ by $(-x_0',-x_1',-x_2')$). Thus, searching through all possible coverings (the total amount is 23040) we find sets $(p_i, m_i, r_i, a_i, b_i)$. Those listed in Table \ref{tab:InformativeTable1} give rise to the smallest initial triple $x_0= 151646890045$, $x_1= 836564809606$, $x_2= 942785024683$, as stated in Theorem \ref{Thm:MainResult}.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\mathbf{i}$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; $10$ &amp; $11$  \\
\hline
$\mathbf{m_i}$ &amp; $2$ &amp; $5$ &amp; $6$ &amp; $8$ &amp; $10$ &amp; $12$ &amp; $15$ &amp; $20$ &amp; $20$ &amp; $24$ &amp; $30$\\ 
\hline
$\mathbf{p_i}$ &amp; $2$ &amp; $29$ &amp; $17$ &amp; $7$ &amp; $11$ &amp; $107$ &amp; $8819$ &amp; $19$ &amp; $239$ &amp; $1151$ &amp; $1621$\\ 
\hline
$\mathbf{r_i}$ &amp; $1$ &amp; $0$ &amp; $4$ &amp; $0$ &amp; $8$ &amp; $8$ &amp; $6$ &amp; $2$ &amp; $14$ &amp; $12$ &amp; $26$\\ 
\hline
$\mathbf{a_i}$ &amp; $1$ &amp; $8$ &amp; $16$ &amp; $3$ &amp; $7$ &amp; $70$ &amp; $3246$ &amp; $12$ &amp; $202$ &amp; $1077$ &amp; $180$\\ 
\hline
$\mathbf{b_i}$ &amp; $0$ &amp; $23$ &amp; $13$ &amp; $1$ &amp; $2$ &amp; $17$ &amp; $8805$ &amp; $8$ &amp; $103$ &amp; $	964$ &amp; $291$\\ 
\hline
\end{tabular}
\end{center}
\caption{\label{tab:InformativeTable1} $p_i, m_i, r_i, a_i, b_i$.}
\end{table}

If we allow non-positive terms in the sequence, the same method gives sets $(p_i', m_i', r_i', a_i', b_i')$, which give the sequence mentioned in the Remark \ref{rem:Rem}: $x_0= 730344594529$, $x_1= -45426674968$, $x_2= 151646890045$ (see Table \ref{tab:InformativeTable2}).

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\mathbf{i}$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; $10$ &amp; $11$  \\
\hline
$\mathbf{m_i}$ &amp; $2$ &amp; $5$ &amp; $6$ &amp; $8$ &amp; $10$ &amp; $12$ &amp; $15$ &amp; $20$ &amp; $20$ &amp; $24$ &amp; $30$\\ 
\hline
$\mathbf{p_i}$ &amp; $2$ &amp; $29$ &amp; $17$ &amp; $7$ &amp; $11$ &amp; $107$ &amp; $8819$ &amp; $19$ &amp; $239$ &amp; $1151$ &amp; $1621$\\ 
\hline
$\mathbf{r_i}$ &amp; $1$ &amp; $2$ &amp; $0$ &amp; $2$ &amp; $0$ &amp; $10$ &amp; $8$ &amp; $4$ &amp; $16$ &amp; $14$ &amp; $28$\\ 
\hline
$\mathbf{a_i}$ &amp; $1$ &amp; $8$ &amp; $7$ &amp; $3$ &amp; $7$ &amp; $70$ &amp; $3246$ &amp; $12$ &amp; $202$ &amp; $1077$ &amp; $180$\\ 
\hline
$\mathbf{b_i}$ &amp; $0$ &amp; $23$ &amp; $11$ &amp; $1$ &amp; $2$ &amp; $17$ &amp; $8805$ &amp; $8$ &amp; $103$ &amp; $964$ &amp; $291$\\ 
\hline
\end{tabular}
\end{center}
  \caption{\label{tab:InformativeTable2} $p_i', m_i', r_i', a_i', b_i'$.}
\end{table}

It is worth noting that in the sequence mentioned in Remark
\ref{rem:Rem} $x_2= 151646890045$, $x_3= 836564809606$, $x_4=
942785024683$, so this means that the sequence in Theorem
\ref{Thm:MainResult}  is a shift of the sequence in Remark
\ref{rem:Rem}.

\end{proof}

Both sequences can be extended to the left. It can be shown that in
both cases mentioned above these extended sequences also contain no
primes. Since the sequences modulo  $P$ are periodic with period
$\lcm(m_1,\ldots,m_{11})=120$, it is enough to check that $x_j\neq k$
modulo $P$, $-8819\leq k\leq 8819=\max(p_i)$, $j=0,\ldots,119$.

\begin{thebibliography}{13}

\bibitem{Dubickas} A.~Dubickas, A.~Novikas, and J.~\v{S}iurys,
\newblock A binary linear recurrence sequence of composite numbers.
\newblock {\em J. Number Theory} \textbf{130} (2010), 1737--1749.

\bibitem{Graham} R.~L. Graham,  \newblock
A Fibonacci-like sequence of composite numbers.  
\newblock {\em Math. Mag.} \textbf{37} (1964), 322--324.
	
\bibitem{Ismailescu} D.~Ismailescu and J.~Son,  \newblock A new kind of 
Fibonacci-like sequence of composite numbers.  \newblock {\em J. Integer Sequences}
\textbf{17} (2014), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL17/Ismailescu/ism8.html}{Article 14.8.2}.

\bibitem{Jaeschke} G.~Jaeschke,  
\newblock On the smallest $k$ such
that all $k2^N + 1$ are composite.
\newblock {\em Math. Comp.} \textbf{40} (1983), 381--384.

\bibitem{Knuth} D.~E. Knuth,  \newblock A Fibonacci-like sequence 
of composite numbers.  \newblock {\em Math. Mag. }
\textbf{63} (1990), 21--25.

\bibitem{Lenstra} A.~K. Lenstra, H.~W. Lenstra, Jr., and L.
~Lov\'{a}sz,
\newblock Factoring polynomials with rational
coefficients.
\newblock {\em Math. Ann.} \textbf{261} (1982), 515--534.

\bibitem{Nicol} J.~W. Nicol,  \newblock A Fibonacci-like sequence of 
composite numbers.  \newblock {\em Electron. J. Combin.} \textbf{6} (1999), \#R44.

\bibitem{Sierpinski} W.~Sierpi\'{n}ski, \newblock Sur un probl\`{e}me concernant les nombres $k2^n+1$. 
\newblock {\em Elem. Math.} \textbf{15} (1960), 63--74.

\bibitem{Siurys1} J.~\v{S}iurys,  \newblock A linear recurrence sequence of composite numbers.  
\newblock {\em LMS J. Comput. Math.} \textbf{15} (2012), 360--373.

\bibitem{Siurys} J.~\v{S}iurys,  \newblock A Tribonacci-like sequence 
of composite numbers.
\newblock {\em Fibonacci Quart.} \textbf{49} (2011), 298--302.

\bibitem{Somer} L.~Somer,  \newblock Second-order linear recurrences of composite numbers.  
\newblock {\em Fibonacci Quart.} \textbf{44} (2006), 358--361.	

\bibitem{Vsemirnov} M.~Vsemirnov,  \newblock A new Fibonacci-like sequence 
of composite numbers.  \newblock {\em J. Integer Sequences}
 \textbf{7} (2004), \href{https://cs.uwaterloo.ca/journals/JIS/VOL7/Vsemirnov/vsem5.html}{Article 04.3.7}.

\bibitem{Wilf} H.~S. Wilf,  \newblock
Letters to the editor.  \newblock {\em Math. Mag. } \textbf{63} (1990), 284.	

\end{thebibliography}


\bigskip
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39; Secondary 11B37.

\noindent \emph{Keywords: } Tribonacci-like recurrence, composite number.

\bigskip
\hrule
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\noindent (Concerned with \seqnum{A000045},
\seqnum{A000073}, and
\seqnum{A056993}.)

\bigskip
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\vspace*{+.1in}
\noindent
Received November 20 2016; 
revised versions received  November 25 2016; December 27 2016;
January 3 2017.
Published in {\it Journal of Integer Sequences}, January 3 2017.

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