Open path theory: Pattern and structure in prime numbers

Abstract

The Open Path theory, supported by experimental data, is presented. The main hypothesis proposes that Prime Numbers's positions are determined by previous Prime Numbers as well as their spacing, in a complex, but deterministic way. The concepts of Open Path, Perfect Space, and Primorial Perfect Space are introduced. The Open Path theory can predict prime gaps of any minimum predetermined size. Two rudimentary algorithms based on this theory are presented. The first algorithm returns a sample (a few hundredth of numbers) containing 25 % of Prime Numbers at distances above 1011. The mirrored sample gives a similar percentage of Prime Numbers. The algorithm execution time is of a few milliseconds. The second algorithm presented determines if a number belonging to a Perfect Space is a composite number or a Prime Number.