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arXiv · 1508.05748

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

Vladimir Shevelev
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Abstract

We study the number $ν(n)$ 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 $n\equiv\pm3 \pmod9,$ $ν(n)>=1;$ (ii) for the exceptional $n,$ $ν(n)=0;$ (iii) for every prime $p\neq3,$ $ν(p)=ν(2p)=1;$ (iv) $\limsup (ν(n))=\infty;$ (v) for every positive $n,$ there exists $k$ such that $ν(k)=n.$

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Vladimir Shevelev. 2016-04-25. Representation of positive integers by the form $x^3+y^3+z^3-3xyz$. https://arxiv.org/abs/1508.05748

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