A simple information theoretical proof of the Fueter-P\'olya Conjecture

TL;DR

This paper provides an information theoretical proof that polynomial pairing functions of degree higher than 2 cannot bijectively map natural numbers to their squares, confirming the Fueter-Pólya Conjecture.

Contribution

It introduces the concept of information efficiency to prove the non-existence of higher degree polynomial pairing functions for natural numbers.

Findings

Cantor functions are information efficient.

Higher order functions cannot be more efficient than Cantor functions.

Polynomial pairing functions of degree > 2 do not exist.

Abstract

We present a simple information theoretical proof of the Fueter-P\'olya Conjecture: there is no polynomial pairing function that defines a bijection between the set of natural numbers N and its product set N^2 of degree higher than 2. We introduce the concept of information efficiency of a function as the balance between the information in the input and the output. We show that 1) Any function defining a computable bijection between an infinite set and the set of natural numbers is information efficient, 2) the Cantor functions satisfy this condition, 3) any hypothetical higher order function defining such a bijection also will be information efficient, i.e. it stays asymtotically close to the Cantor functions and thus cannot be a higher order function.