Cantor polynomials and the Fueter-Polya theorem

TL;DR

This paper presents an elementary proof of the Fueter-Polya theorem, which states that the only quadratic packing polynomials are the Cantor polynomials, resolving a century-old conjecture in lattice point enumeration.

Contribution

It provides a new, elementary proof of the Fueter-Polya theorem, simplifying the understanding of quadratic packing polynomials.

Findings

Confirmed the uniqueness of Cantor polynomials as quadratic packing polynomials

Provided an elementary proof of the Fueter-Polya theorem

Resolved a century-old conjecture in the theory of packing polynomials

Abstract

A packing polynomial is a polynomial that maps the set $\mathbf{N}_0^2$ of lattice points with nonnegative coordinates bijectively onto $\mathbf{N}_0$. Cantor constructed two quadratic packing polynomials, and Fueter and Polya proved analytically that the Cantor polynomials are the only quadratic packing polynomials. The purpose of this paper is to present a beautiful elementary proof of Vsemirnov of the Fueter-Polya theorem. It is a century-old conjecture that t