Jacob’s Ladder: Prime Numbers in 2D

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 employ that to 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 property: there is a clear exponential decay in the frequency of the gaps vs gap size. The smaller the gaps, the more frequently they appear.

A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories

“…In [19], a simple representation of prime numbers in 2D is introduced that may lead to important avenues in the field of research on prime numbers. The proposed numerical representation yielded an interesting visual structure that depicts an oscillating plot, which ascends and descends according to the distribution of prime numbers.…”

“…On a small scale, the distribution of prime numbers appears random; however, when considered on a large scale, it appears to have a pattern that is not fully understood. This interplay between regular and chaotic behaviors has motivated researchers for a detailed study in this area that could possibly be signatures of more fundamental mathematical properties and features [19]. The statistical properties of prime numbers are one way of investigating their behavior and characteristics.…”

Persistent order in Schramm-Loewner Evolution driven by the primes last digit sequence

“…In previous work [1], [2] a set of new geometrical methods were proposed for the study of the primes numbers distribution and their associated last digits sequence resulting from taking their (mod10) congruences. The latter sequence is known to be equivalent to a quaternary symbolic sequence and has been shown in [2] to have some unusual characteristics when projecting it as a 2D random walk via a special binarization filter.…”