Number Theories

TL;DR

This paper extends fundamental number theory concepts to arbitrary operations, introducing generalized distributivity, primes, and modulo arithmetic, which offers new perspectives on classical problems like the Goldbach conjecture.

Contribution

It develops a generalized framework for number theory concepts applicable to arbitrary operations, including hyperoperations and n-ary functions, and explores their implications.

Findings

Generalized distributivity for hyperoperations

A new definition of primes for arbitrary operations

Connections between factorization and the Riemann zeta function

Abstract

We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization of the fundamental theorem of arithmetic for hyperoperations. We also give a generalized definition of the prime numbers that are associated to an arbitrary n-ary operation and take a few steps toward the development of its modulo arithmetic by investigating a generalized form of Fermat's little theorem. Those constructions give an interesting way to interpret diophantine equations and we will see that the uniqueness of factorization under an arbitrary operation can be linked with the Riemann zeta function. This language of generalized primes and composites can be used to restate and extend certain problems such as the Goldbach conjecture.