Jacob's Ladder: Prime numbers in 2d

TL;DR

This paper introduces a 2D representation of prime numbers that reveals new conjectures and properties, including exponential decay in gap frequencies and an approximate prime count within the sequence, offering fresh insights into prime distribution.

Contribution

The work presents a novel 2D visualization of primes that formulates new conjectures and uncovers properties not previously observed, advancing prime number research.

Findings

Exponential decay in gap frequency vs. gap size.

Sequence of zeroes contains approximately n/log(n) primes.

Zeroes grow in an erratic but patterned manner.

Abstract

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them present a remarkable and intriguing property: a clear exponential decay in the frequency of gaps vs gap size. The smaller the gaps, the more frequently they appear. Additionally, the sequence of zeroes, despite being non-consecutive numbers, contains a number of primes approximately equal to n/log(n) , being n the number of terms in the sequence.