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

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 $\nu(k)=n.$