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

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.