A simple information theoretical proof of the Fueter-Pólya Conjecture

09/26/2018
by   Pieter W. Adriaans, et al.
0

We present a simple information theoretical proof of the Fueter-Pólya 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/16/2023

The sequence of higher order Mersenne numbers and associated binomial transforms

In this article, we introduce and study a new integer sequence referred ...
research
12/19/2019

Hyperpfaffians and Geometric Complexity Theory

The hyperpfaffian polynomial was introduced by Barvinok in 1995 as a nat...
research
11/25/2019

BiEntropy, TriEntropy and Primality

The order and disorder of binary representations of the natural numbers ...
research
10/28/2020

A Cyclic Proof System for HFLN

A cyclic proof system allows us to perform inductive reasoning without e...
research
08/29/2021

A new proof of the Gasca-Maeztu conjecture for n = 5

An n-correct node set 𝒳 is called GC_n set if the fundamental polynomial...
research
07/09/2021

Higher Order Imprecise Probabilities and Statistical Testing

We generalize standard credal set models for imprecise probabilities to ...
research
11/08/2021

Differential information theory

This paper presents a new foundational approach to information theory ba...

Please sign up or login with your details

Forgot password? Click here to reset