# A conjecture about special linear group and Fibonacci number

Definition: $T_m=\{A\in M_{m\times \infty}(\Bbb{Z})\mid a_{ij}\cdot a_{i(j+1)}-a_{(i-1)(j+1)}\cdot a_{(i+1)j}=1, \forall i\geq 2, j\geq 1, a_{1j}=1=a_{mj}, \forall j=1,2,...\}$.

Claim: $\forall m\in \Bbb{Z}^+$, $T_m\neq \emptyset$.

Claim: $\forall m\in \Bbb{Z}^+$, $\#T_m<\infty$.

Claim: $\forall m\in \Bbb{Z}^+$, $\forall A\in T_m$, column of $A$ has period $p(A)$.

Conjecture: $$g(m)=\max_{k=1}^{\# T_m}\min\{\max A_k c_1, \max A_k c_2, \max A_k c_3,..., \max A_k c_{p(A_k)}\}$$ is $1,1,2,2,3,3,5,5,8,8,13,13,...$, where $c_j$ is the column vector which has $1$ at the $j$th position and $0$ elsewhere.

This conjecture comes from the following filling problem. $\begin{pmatrix} 1 & & 1 & & 1 & & 1 & \cdots\\ & a & & b && c&& d & \cdots\\ & & 1 & & 1 & & 1 & & 1 & \cdots\\ \end{pmatrix}$,
$a, b, c, d, ...$ should satisfy
$ab-1\cdot 1=1$, $bc-1\cdot 1=1$, $cd-1\cdot 1=1$, ...

$\begin{pmatrix} 1 & & 1 & & 1 & & 1 & \cdots\\ & a & & b && c&& d & \cdots\\ & & \alpha & & \beta && \gamma && \delta & \cdots\\ & & & 1 & & 1 & & 1 & & 1 & \cdots\\ \end{pmatrix}$,
$a, b, c, d, ...$, $\alpha, \beta, \gamma, \delta, ...$ should satisfy
$ab-1\cdot \alpha=1$, $bc-1\cdot \beta=1$, $cd-1\cdot \gamma=1$, ...
$\alpha \beta-b=1$, $\beta\gamma-c=1$, $\gamma\delta-d=1$, ...

For example, $A_1=\begin{pmatrix} 1 & & 1 & & 1 & & 1 & \cdots\\ & 1 & & 2 && 1&& 2 & \cdots\\ & & 1 & & 1 & & 1 & & 1 & \cdots\\ \end{pmatrix}\in T_3$, $P(A_1)=2$.

Or

$A_1=\begin{pmatrix} 1 && 1 & & 1 & & 1 && 1 && 1 && 1 &\cdots\\ & 1 && 2 && 2 && 1 && 3 && 1 && 2 & \cdots\\ && 1 && 3 && 1 && 2 && 2 && 1 && 3 & \cdots\\ &&& 1 && 1 && 1 && 1 && 1 && 1 && 1 & \cdots\\ \end{pmatrix}\in T_4$, $P(A_1)=5$.

Write it by matrix language.

It is easy to show that $T_3=\left\{A_1=\begin{pmatrix} 1 & & 1 & & 1 & & 1 & \cdots\\ & 1 & & 2 && 1&& 2 & \cdots\\ & & 1 & & 1 & & 1 & & 1 & \cdots\\ \end{pmatrix}\right\}$

and

$T_4=\left\{A_1=\begin{pmatrix} 1 && 1 & & 1 & & 1 && 1 && 1 && 1 &\cdots\\ & 1 && 2 && 2 && 1 && 3 && 1 && 2 & \cdots\\ && 1 && 3 && 1 && 2 && 2 && 1 && 3 & \cdots\\ &&& 1 && 1 && 1 && 1 && 1 && 1 && 1 & \cdots\\ \end{pmatrix}\right\}$.

This conjecture was given by my professor. He had checked it by computer with $m\leq 10$.

• Hard to see what $T_m$ is... could you include the first few in the question? – coffeemath Apr 26 '17 at 15:02
• @coffeemath Thanks for your comment. – bfhaha Apr 26 '17 at 16:59
• Can you explain a bit more what g(m) is actually keeping track of? The max-min-max definition is difficult to parse. – Nate Apr 26 '17 at 17:10
• One not about your numbers ($1,1,2,2,3,3,...$) is that these numbers can be generated by the sequence $a_n=a_{n-2}+a_{n-4}$ with initial values $a_0=0,a_1=a_2=a_3=1$. Maybe helpful. – Amin235 Apr 27 '17 at 22:06