This paper has used methods from several disciplines to study the behaviour of the nonlinear cellular automaton of Eq. (3.1). Despite the simplicity of its construction, all the approaches taken support the conjecture that its behaviour is so complicated as to seem random for practical purposes. It is remarkable that such a simple system can give rise to such complexity. But it is in keeping with the observation that mathematical systems with few axioms, or computers with few intrinsic instructions, can lead to essentially arbitrary complexity. And it seems likely that the mathematical mechanisms at work are also responsible for much of the randomness and chaos seen in nature.
The simplicity of Eq. (3.1) makes it amenable to highly efficient practical implementation. And the analyses carried out here suggest that the sequences it produces have a high degree of randomness. In fact, if any regularity could be found in these sequences, it would probably have substantial consequences for studies of many complex and seemingly random phenomena.