I arrange the odd numbers starting at $3$ into infinitely many rows by $k$ columns. For example if $n=2$:
$$ A_2 =\begin{matrix}\ \color{red}{3}&5\\ \color{red}{7}&9\\ \color{red}{11}&13\\ 15&\color{red}{17}\\ \color{red}{19}&21\\ \color{red}{23}&25\\ 27&\color{red}{29}\\ \color{red}{31}&33\\ 35&\color{red}{37}\\ 39&\color{red}{41}\\ \color{red}{43}&45\\ \color{red}{47}&49\\ 51&\color{red}{53}\\ 55&57\\ \vdots&\vdots\\ \end{matrix}$$
I then try and see how far down I can walk by only traversing cells with prime numbers. I denote the length of the longest prime walk in $A_k$ by $w_k$. In the above example we have a walk $$w=\{3,7,11,17,19,23,29,31,37,41,43,47,53\}$$ consequently $w_2=13$. Note there may be more than one "longest prime walk". Also back tracking is allowed in order to reach over columns and continue your way down. The table below lists some values by hand:
$$ \begin{array}{c|lcr} \text{# columns }=k &1&2&3&4&5&6&7&8&9&10&11&12&13\\ \hline \text{longest walk }=w_k&3&13&19&14&11&15&22&8&10&26&32&10&19\\ \end{array} $$
I have two simple questions:
- Is it true that $w_k>0$ for every $k$? Can we always take a first step?
- Can we take arbitrarily long walks down $A_k ?$
To me question one is interesting. Is it possible for $k$ to get so large that no matter what prime we start off with on the first row we cannot take a step to the second row. I would imagine this is related to prime gaps particularly when the last number in the first row is a prime. Question two is straightforward: Is there any reason to believe there exist a constant $m$ such that $w_k <m$ for every $k$. The longest walk I have made so far is $32$.
This is just pure curiosity. I would be interested if anyone could generate really long walks.