Fractal Dimension, Primes, and the Persistence of Memory

ABSTRACT . Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales—a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

Download the paper (File: fpm.pdf, about 220 Kb). Appeared in the journal Advances in Complex Systems.

©2002 J. L. Pe. Document created on 31 March 2003.