Divisibility and Cellular Automata

Cellular automata (CA) are perfect feedback machines which change the state of their cells step by step. In a certain sense, Pascal’s triangle was the first CA and there is a strong connection between Pascal’s triangle and the tractal pattern formation known as Sierpinski gasket. Generalizing divisibility properties of the coefficients of Pascal’s triangle, binomial arrays as well as gaussian arrays are evaluated mod p. In these arrays, two fiactal geometric characteristics are evident: a) self-similarity and b) non integer dimension. The conclusions at which we arriveas well as the conjectures we propose, are important facts to take into account when modelhng real experiments like catalytic oxidation reactions in Chemistry, where the remarkable resemblance of the graph: number of entries in the p row of the Pascal’s triangle which are not divisible by 2 vs k and the measurement of the chemical reaction rate as a function of time, provides the reason to model a catalytic converter by a one-dimensional CA [4].