On Cross Hyperoperatorial Migration of Properties, Related to Natural Number Division Operator

TL;DR

This paper extends integer divisibility and prime factor concepts across the entire hyperoperation hierarchy, establishing a new unique representation of natural numbers as tower-like exponentiation structures called biprimes.

Contribution

It introduces a unified framework for divisibility and prime concepts across hyperoperations and formulates an exponentiation-based theorem for natural number representation.

Findings

Defines divisibility and prime concepts across hyperoperations

Establishes a unique tower-like exponentiation representation of natural numbers

Formulates an exponentiation-based fundamental theorem of arithmetic

Abstract

In the article integer divisibility properties and related prime factors natural number representation concepts have been defined over the whole infinite hyperoperation hierarchy. The definitions have been made across and above of unique arithmetic operations, composing this hierarchy (addition, multiplication, exponentiation, tetration and so on). It allows the habitual concepts of "prime factor", "multiplier", "divider", "natural number factors representation" etc., to be associated mainly with the same sense, with the each of those operations. As analogy of multiplication-based Fundamental Theorem of Arithmetic (FTA), an exponentiation-based theorem is formulated. The theorem states that any natural number $M$ can be uniquely represented as a tower-like exponentiation: $M=a_n\uparrow(a_{n-1}\uparrow (\ldots (a_2\uparrow a_1)\ldots )),$ where $a_i\neq 1 (i=1,\ldots,n)$ are primitive in some sense (related to the exponentiation operation), exponentiation components, following one by one in some unique order and named in the article as biprimes.