Representation of positive integers by the form $x^3+y^3+z^3-3xyz$

TL;DR

This paper investigates the representation of positive integers by a specific cubic form, establishing conditions for existence, uniqueness, and the behavior of the number of representations, revealing both limitations and infinite variability.

Contribution

It provides a comprehensive analysis of the representation counts for the form, including residue conditions, prime cases, and the unbounded nature of the number of representations.

Findings

All positive integers except those congruent to ±3 mod 9 are represented.

For primes p ≠ 3, the form represents p and 2p exactly once.

The maximum number of representations for some integers is unbounded.

Abstract

We study the number $\nu(n)$ of representations of a positive integer $n$ by the form $x^3+y^3+z^3-3xyz$ in the conditions $0\leq x\leq y\leq z; z\geq x+1.$ We proved the following results: (i) for every positive $n,$ except for $n\equiv\pm3 \pmod9,$ $\nu(n)>=1;$ (ii) for the exceptional $n,$ $\nu(n)=0;$ (iii) for every prime $p\neq3,$ $\nu(p)=\nu(2p)=1;$ (iv) $\limsup (\nu(n))=\infty;$ (v) for every positive $n,$ there exists $k$ such that $\nu(k)=n.$