Such cycles are manifest as closed loops in the state transition graph. Here is a number theoretical function defined to be the minimum positive integer for which modulo . The maximum value of , typically achieved when is prime, is . The maximal cycle length is thus of order , approximately the square root of the total number of possible states . An unusual feature of this analysis is the appearance of number theoretical concepts.
Number theory is inundated with complex results based on very simple premises.
It may be part of the mathematical mechanism by which natural systems of simple construction yield complex behavior.