Notes on properties of binary strings which encode prime occurrences

TL;DR

This paper explores binary string representations of prime occurrences, introduces three algorithms for generating these strings, and demonstrates their use in analyzing prime distribution and related conjectures.

Contribution

It presents three simple algorithms for generating binary strings encoding prime positions and applies them to prime distribution analysis and Goldbach's hypothesis.

Findings

Algorithms successfully generate prime occurrence strings

Application to limited Goldbach's hypothesis cases

Formulation of three open conjectures about prime distribution

Abstract

A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a broad range of existing string-analysis algorithms to problems in number theory. Binary strings of prime occurrences can be also generated with simple algorithms. Here we discuss three such algorithms and we demonstrate their applicability using the example of proving Goldbach's hypothesis for some limited sets of even numbers. This work formulates three open questions (conjectures) regarding the distribution of primes.