A Natural Partial Order on The Prime Numbers

TL;DR

This paper introduces a natural partial order on prime numbers based on internal symmetries of finite fields, linking it to Hopf algebras and offering a new perspective on the Prime Number Theorem.

Contribution

It presents a novel partial order on primes derived from finite field symmetries, connecting algebraic structures to number theory and primality testing.

Findings

Establishes a correspondence between prime numbers and rooted trees

Provides an alternative approach to the Prime Number Theorem

Links finite field symmetries with Hopf algebra structures

Abstract

A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.