A recursive function coding number theoretic functions

TL;DR

The paper demonstrates the existence of a fixed recursive function that can encode any number-theoretic function through a specific conjugation, revealing a universal recursive coding scheme.

Contribution

It introduces a universal recursive function that can represent any number-theoretic function via an injective conjugation, advancing the understanding of recursive function coding.

Findings

Existence of a fixed recursive function e with universal coding properties

Construction of injective functions c_h for all h

Representation of any h as conjugate of e via c_h

Abstract

We show that there exists a fixed recursive function $e$ such that for all functions $h\colon \mathbb{N}\to \mathbb{N}$, there exists an injective function $c_h\colon \mathbb{N}\to \mathbb{N}$ such that $c_h(h(n))=e(c_h(n))$, i.e., $h=c_h^{-1}ec_h$.