  The Kauffman model of genetic computation highlights the importance of
criticality at the border of order and chaos. The model with connectivity one
is of special interest because it is exactly solvable. But our understanding of
its behavior is incomplete, and much of what we do know relies on heuristic
arguments. Here, we show that the key quantities in the model are intimately
related to aspects of number theory. Using these links, we derive improved
bounds for the number of attractors as well as the mean attractor length, which
is harder to compute. Our work suggests that number theory is the natural
language for deducing many properties of the critical Kauffman model with
connectivity one, and opens the door to further insight into this deceptively
simple model.
