On the statistical distribution of prime numbers, a view from where the distribution of prime numbers is not erratic

TL;DR

This paper explores the distribution of prime numbers, proposing a new characterization using a specific function to distinguish primes from non-primes among odd numbers, and discusses their properties in relation to classical number theory results.

Contribution

It introduces a novel function-based characterization of prime numbers among odd natural numbers, offering a new perspective on their distribution and properties.

Findings

The function 2ab+a+b can distinguish primes from non-primes among odd numbers.

Discussion of prime distribution in relation to the prime number theorem and twin primes.

Insights into the erratic nature of prime distribution and connections to Fermat and Euler numbers.

Abstract

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.