On algebraic structure of the set of prime numbers

TL;DR

This paper explores the algebraic and arithmetical structure of prime numbers, proposing a formula that redefines primes as combinations of natural number subsets, similar to formulas for odd numbers, and extends to composites.

Contribution

It introduces a new algebraic formula for prime numbers and demonstrates how primes and composites can be characterized through set operations and primary structures.

Findings

Prime numbers can be expressed as unions and intersections of natural number subsets.

The proposed formula for primes is analogous to the 2n - 1 formula for odd numbers.

All composite numbers can be defined using the derived prime formula.

Abstract

The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has been shown that the set of prime numbers is combinations (unions and intersections) of some subsets of natural numbers, with more primary structures. In fact generally, the logical essence of obtained formula for prime numbers is similar to formula 2n - 1 for odd numbers, and so on. Subsequently, using obtained formula we can define all composite numbers. Finally specified examples for obtained formula are presented.